LIBRARY  * 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

/ 
Class 


HIGH  MASONRY  DAM 
DESIGN 


BY 

CHARLES  E.   MORRISON,   C.E.,   PH.D. 

AssocV<Mem.  Am.  Soc.  C.  E. 
AND 

ORRIN  L.  BRODIE,  C.E. 

Assoc.  Mem.  Am.  Soc.  C.  E.,  Asst.  Eng.  Board  Water  Supply,  N.  Y.  City 


FIRST    EDITION 
FIRST    THOUSAND 


NEW  YORK 

JOHN  WILEY  &   SONS 
LONDON:    CHAPMAN  &  HALL,   LIMITED 


1910 


A 


Copyright,  1910 


i  BY 

CHARLES  E.  MORRISON    AND    ORRIN  L.  BRODIE 


\]t  &rirntifit  Press 
^Brttmtnonb  and 


PREFACE 


IT  is  the  practice  at  Columbia  University  to  require  of 
the  third-year  students  in  the  Department  of  Civil  Engineer- 
ing, the  execution  of  the  design  of  a  masonry  dam,  and 
to  aid  them  in  this  problem  they  have  heretofore  been 
furnished  with  "Notes  on  the  Theory  and  Design  of  High 
Masonry  Dams,"  prepared  some  years  ago  by  Prof.  Burr 
of  the  Department,  and  having  for  their  basis  the  method 
as  set  forth  by  Mr.  Edward  Wegmann. 

This  procedure  with  which  Wegmann  is  credited,  and 
which  was  developed  through  the  investigations  undertaken 
in  connection  with  the  Aqueduct  Commission  of  the  city 
of  New  York,  for  the  purpose  of  determining  a  correct 
cross-section  for  the  Quaker  Bridge  dam,  resulted  in 
the  first  direct  method  for  calculating  the  cross-section 
of  such  structures  and  is  essentially  a  development  of 
the  Rankine  theory. 

The  studies  appeared  first  in  the  report  made  by  Mr. 
A.  Fteley  to  the  chief  engineer  of  the  Aqueduct  Com- 
mission of  the  city  of  New  York,  dated  July  25,  1887, 
and  later  in  Mr.  Wegmann's  treatise  on  "The  Design 
and  Construction  of  Dams." 

Neither  in  the  report  nor  in  the  treatise  however, 
have  the  effects  of  uplift,  due  to  water  permeating  the 


201338 


iv  PREFACE 

mass  of  masonry,  and  of  ice  thrust,  acting  at  the  sur- 
face of  the  water  in  the  reservoir,  been  considered,  and 
in  consequence  of  this,  objection  might  be  legitimately 
raised  that  the  series  of  equations  determining  the  cross- 
section  fails  to  account  for  these  factors.  Some  difference 
of  opinion  may  exist  as  to  the  relative  importance  of 
these  considerations,  but  when  a  structure  of  great 
responsibility  is  projected,  conservatism  in  design  is 
essential. 

The  following  presentation  which  aims  to  supply  these 
omissions,  has  been  prepared  primarily  that  there  may  be 
had  in  convenient  form  a  text,  containing  the  general 
treatment  and  such  consideration  of  these  factors  as  more 
recent  practice  requires,  together  with  a  brief  statement 
regarding  the  late  investigations  undertaken  for  the  pur- 
pose of  determining  more  accurately  the  variation  of 
stress  in  masonry  dams. 

The  formulae  relating  to  uplift,  ice  thrust,  etc.,  were 
deduced  by  one  of  the  authors  and  have  been  used  in  part 
in  connection  with  the  design  of  the  large  dams  for  the 
new  water  supply  for  the  city  of  New  York. 

The  computations  for  the  design  of  a  high  masonry 
dam  are  appended  to  facilitate  the  ready  comprehension 
and  application  of  the  formulae. 

It  is  hoped  that  the  presentation  may  appeal  to  the 
practicing  engineer  as  well  as  the  student,  and  that  there 
may  be  found  the'rein  enough  to  compensate  him  for  the 

labor  involved  in  its  perusal. 

C.  E.  M. 

O.  L.  B. 

COLUMBIA  UNIVERSITY,  1910. 


OF   THE 

UNIVERSITY 

OF 


HIGH  MASONRY  DAM  DESIGN 


THE  method  of  analysis  by  which  an  economical  cross- 
section  of  a  high  masonry  dam  may  be  most  directly 
calculated,  and  the  one  which  is  most  generally  adopted 
in  engineering  practice,  was  first  devised  by  Mr.  Edward 
Wegmann  through  studies  made  in  connection  with  the 
Aqueduct  Commissioners  of  New  York  city,  and  it  is 
that  method  which  will  be  employed  here,  though  it  will 
receive  some  modification  in  certain  particulars  and  be 
elaborated  in  certain  others. 

In  determining  the  cross-section  by  the  series  of  equa- 
tions developed  in  that  analysis,  no  account  is  taken  of 
the  condition  of  uplift  due  to  water  penetrating  the  mass 
of  masonry,  nor  of  the  ice  thrust  acting  horizontally  at 
the  surface  of  the  water  in  the  reservoir  against  the  up- 
stream face  of  the  dam,  though  reference  is  made  to  it. 
Present  practice  requires  however,  that  these  two  factors 
be  considered  where  a  structure  of  great  responsibility 
is  proposed,  and  in  this  respect  at  least,  will  the  analysis 
be  amplified. 

While  upward  pressure  in  a  masonry  dam,  either 
at  the  foundation  or  in  joints  higher  up,  should  always 
be  considered,  the  subject  does  not  lend  itself  to  a  very 


2  HIGH    MASONRY    DAM    DESIGN 

exact  treatment;  in  fact,  it  becomes  necessary  to  make 
assumptions  in  regard  to  its  presence  and  action  which 
in  the  end  depend  principally  upon  the  judgment  of  the 
engineer.  It  is  not  surprising  therefore,  that  a  wide 
range  of  opinion  exists  as  to  the  method  of  dealing  with 
this  factor. 

Such  pressures  may  become  effective  from  two  causes: 
the  percolation  of  water  into  small  cracks  either  in  the 
superstructure  or  the  foundation,  or  by  the  presence  of 
springs  in  the  foundation  itself. 

In  the  best  laid  masonry  it  is  undoubtedly  true  that 
small  cracks  exist  into  which  the  water  gains  entrance, 
but  this  should  be  guarded  against  as  far  as  possible  by 
the  exercise  of  great  care  in  the  laying  of  the  stone  and 
the  bonding  of  them,  together  with  thorough  inspection. 
"Temperature  variations  due  to  setting  of  concrete  and 
also  due  to  daily  and  seasonal  changes,  while  inducing 
stresses  that  are  indeterminate,  thereby  providing  an 
argument  for  conservatism  in  design,  in  addition  affect 
permeability  to  greater  or  less  degree."  Even  in  cyclo- 
pean  masonry  where  no  horizontal  joints  exist,  except 
between  the  facing  stones,  the  possibility  of  other  small 
cracks  being  formed  is  always  present,  and  therefore 
requires  recognition  here  as  well. 

Where  springs  are  found  in  the  native  rock  of  the 
foundation  the  effect  of  upward  pressure  from  such  a 
source  must  be  overcome  by  a  system  of  drains  which 
shall  lead  the  water  below  the  downstream  face  of  the  dam. 
The  difficulty  here  is  from  the  possible  formation  of  new 
springs  as  soon  as  the  reservoir  becomes  filled,  with  which 
some  connection  will  inevitably  be  formed,  and  thus 


HIGH    MASONRY    DAM    DESIGN  3 

cause  an  upward  pressure  due  to  the  total  hydrostatic 
head  of  the  water  back  of  the  dam.  It  is  evident  there- 
fore, that  the  foundation  should  be  carefully  examined 
and  be  specially  prepared  to  receive  the  first  course  of 
masonry. 

The  method  of  allowing  for  upward  pressure  depends 
upon  its  properly  assumed  presence,  and  also  upon  some 
assumed  law  of  variation.  Present  practice  indicates 
that  this  may  be  considered  as  varying  from  a  maximum 
at  the  heel  to  zero  at  the  toe.  But  because  it  is  hardly 
justified  from  experience  with  other  dams  to  impose  such 
severe  conditions  upon  the  masonry  above,  it  is  agreed 
to  consider  this  pressure  as  acting  over  only  a  portion 
of  the  joint,  or,  in  other  words,  to  consider  only  a  portion 
of  the  full  hydrostatic  head  as  acting  at  the  upstream 
face  of  the  joint  under  examination. 

Although  lack  of  exact  data  precludes  the  possibility 
of  assigning  a  definite  value  to  the  force  of  expanding 
ice  in  its  formation  at  the  surface  of  a  reservoir,  yet  it 
will  be  evident  that  a  thrust  from  such  a  cause  may 
greatly  affect  the  dimensions  of  a  profile,  in  cold  climates 
especially.  As  this  thrust  is  effective  at  the  surface  of 
the  water,  for  a  low  structure  it  may  become  a  very  serious 
feature. 

The  studies  involved  in  the  determination  of  a  cross- 
section  demand  an  investigation  along  two  general  lines: 

First,  the  direct  calculation  fixing  the  most  economical 
cross-section  under  the  imposed  conditions,  and 

Second,  studies  in  comparing  cross-sections  ranging 
between  this  one,  which  may  be  called  the  minimum, 
and  one  of  an  existing  masonry  dam,  where  the  conditions 


4  HIGH    MASONRY    DAM    DESIGN 

and  responsibility  are  practically  the  same  as  those  undei 
consideration.  Before  undertaking  such  an  analysis  how- 
ever, it  will  be  necessary  to  consider  the  manner  in  which 
water  pressure  is  exerted  against  a  submerged  surface; 
its  amount;  the  method  of  determining  the  point  of  ap- 
plication of  the  resultant;  the  assumed  distribution  of 
pressure  in  a  masonry  joint;  and  finally,  the  action  of  the 
forces  in  and  upon  the  structure. 

It  may  be  stated  as  a  general  proposition  that  water 
pressure  acts  in  all  directions  against  a  submerged  object 
and  that  it  depends  for  its  value  merely  upon  the  "  head," 
or  depth  of  the  center  of  gravity  of  the  figure  below  the 
free  surface  of  the  liquid.  In  consequence  of  this  principle 
it  may  be  shown  that  the  total  normal  pressure  is  repre- 
sented by 

.......    (i) 


where  P  =  the  total  normal  pressure  ; 

?-  =  the  weight  of  a  unit  volume  of  water; 
A  =the  total  area;  and 

h  =  the  vertical   distance   of   the   figure's   center   of 
gravity  beneath  the  free  surface  of  water. 

The  demonstration*  may  be  made  by  considering  the 
surface  divided  into  an  infinite  number  of  parts;  the  total 
pressure  on  each  one  of  these  elements,  depending  only 
upon  the  weight  of  water  resting  upon  it,  may  be  written, 


(2) 


*  See  Merriman's  "  Hydraulics." 


HIGH    MASONRY    DAM    DESIGN  5 

in  which  p  =  the  total  normal  pressure  on  the  differential 

area; 

a  =  the  differential  area ; 

hi  =  the  head  on  a  (practically  constant  over  the 
differential  area). 

If,  therefore,  we  take  the  sum  of  the  pressures  on  all  of 
these  small  areas,  we  shall  obtain  the  previous  equation, 
which  is  perfectly  general  and  applies  to  any  surface. 
In  the  case  of  a  vertical,  rectangular  strip  of  the  back 
of  a  dam,  the  application  of  the  formula  will  give  a  total 
pressure  of 

(3) 

where  b  is  the  constant  breadth  of  the  strip,  usually  taken 
as  one  unit,  x  is  the  variable,  and  H  is  the  total  height 
of  the  rectangle. 

The  point  on  the  submerged  surface  at  which  this 
resultant  pressure  acts  may  be  determined  by  assuming 
arbitrarily  any  axis,  taking  the  moment  of  inertia  of  the 
surface  about  this  axis,  and  dividing  the  result  by  the 
static  moment  of  the  surface  with  reference  to  the  same 
axis.  Applying  this  to  the  strip  referred  to  above  and 
assuming  the  arbitrary  axis  to  be  the  horizontal  line  in 
which  the  surface  of  the  water  cuts  the  plane  of  the  back, 
there  will  result, 


6ff3       bH3       bHZ 

12         4          3    ' 


6  HIGH   MASONRY    DAM    DESIGN 

which  is  the  moment  of   inertia  with   regard   to  the  as 
sumed  axis. 

H    bH* 


is  the  static  moment  of  the  surface  about  the  same  axis, 
hence, 

7,     *H 


is  the  distance  of  the  center  of  pressure  from  the  surface 
of  the  water. 

In  the  investigation  of  the  distribution  of  pressure  in 
a  masonry  joint  subjected  to  external  forces,  the  material 
is  assumed  to  be  rigid,  though  in  reality  it  is  to  a  certain 
degree-  elastic.  This  elasticity  gives  the  distribution  of 
stress  an  indeterminate  law,  so  that  neither  the  direction 
nor  the  intensity  is  actually  known  at  any  point.  It  is 
certain,  however,  that  the  intensity  must  be  zero  at  the 
edges,  although  it  may  increase  with  great  rapidity  to 
higher  values  very  near  the  limits  of  the  joint.  Investi- 
gations have  been  made  within  the  past  few  years  to 
obtain  more  exact  information  as  to  this  distribution  of 
stress,  but  so  far  the  results  are  not  completely  satisfac- 
tory. Reference  will  be  made  to  this  matter  in  the  Ap- 
pendix. 

Inasmuch  as  the  exact  law  of  stress  variation  is  not 
known,  one  of  uniform  variation  of  normal  stress  has  been 
assumed  in  all  practical  treatments  of  masonry  joints. 

Fig.  i  represents  the  simplest  case,  in  which  the 
pressure  is  assumed  to  be  uniformly  distributed  over  the 


HIGH    MASONRY    DAM    DESIGN  7 

joint  a  6,  with  the  constant  intensity  p\  it  might  be  taken 
as  representing  any  horizontal  joint  with  a  superimposed 
load  acting  at  its  center. 

To  express  this  condition  of  uniform  stress  algebra- 
ically, /  may  be  assumed  to  be  the  length  of  the  joint 
from  a  .to  b,  while  the  breadth,  perpendicular  to  the  plane 
of  the  paper,  is  taken  as  unity.  The  area  of  the  joint 
will  then  be  /,  whence, 

W-pl, (5) 

or 

W 
P~T> (6) 

the  uniform  intensity  of  stress  over  the  entire  joint. 

It  may  be  observed  here  that  this  pressure  is  uniform 
only  because  the  total  load  represented  by  W,  acts  at 
the  center  of  the  joint,  and  that 
when  the  point  of  application  is 
changed  to  some  other  position, 
there  will  be  an  increased  stress 
in  that  direction  toward  which 

the  load  has  been   moved,  and 

t 
a  corresponding  decrease  in  the 

FIG.  i. 
opposite  direction. 

It  will  be  necessary  therefore,  to  consider  this  varia- 
tion of  pressure  in  eccentrically  loaded  joints  and  also 
the  manner  in  which  the  eccentricity  in  the  case  of  a  dam 
is  produced. 

If  a  b  be  any  plane,  horizontal  joint  in  the  dam  at  the 
distance  H  below  the  surface,  OY  the  water  surface,  and 
<£  the  angle  that  the  back  makes  with  the  vertical,  then 


,4 


HIGH    MASONRY    DAM    DESIGN 


the  total  pressure  on  the  back  acting  at  a  point  one-  third 
the  distance  up,  will  be 


Combining  this  force  with  the  weight  of  masonry  W 
above  the  joint  acting  through  the  center  of  gravity  of  the 
section,  the  resultant  R  will  intersect  it  at  some  point  as 
e,  on  a  b,  other  than  the  center  of  figure,  called  the  center 
of  resistance,  and  it  is  evident  that  with  a  variation  of 
Ff  and  W  it  may  occupy  any  position  along  the  joint. 

Fig.  2,  showing  only  the  vertical  component,  exhibits 
such  a  case,  where  compression  exists  over  the  entire  joint 

as  in  Fig.  i,  but  where  the 
center  of  pressure  is  not  at  the 
center  of  figure. 

If  the  intensity  of  pressure 
at  b  may  be  represented  by  the 
vertical  line  p,  and  the  intensity 
of  pressure  at  a  by  the  line  p', 
then,  since  by  the  fundamental 
assumption  the  pressure  varies  uniformly  over  the  entire 
joint,  the  vertical  at  any  point,  included  between  the 
horizontal  a  b  and  the  line  joining  the  extremities  of  p 
and  pr  will  indicate  the  intensity  of  pressure  at  that 
point,  while  the  area  of  the  trapezoid  will  represent  the 
total  pressure  on  the  joint. 

The  former  may  be  expressed  algebraically  thus: 


(7) 


HIGH    MASONRY    DAM    DESIGN  9 

and  the  latter  by, 

i  Jh  4-  Jh'\ 

(8) 

The  determination  of  the  maximum  and  minimum 
pressure  p  and  p'  may  be  made  as  follows : 

Since  the  static  moment  of  the  rectangle  p  I  about  a 
point  JZ  from  pr  is  the  same  as  the  static  moment  of  the 
trapezoid  about  the  same  point,  because  the  moment  of 
the  triangle  p— p',  I  about  that  point  is  zero,  that  being 
the  center  of  gravity  of  the  triangle,  there  will  result  by 
taking  moments 

_pp 

Q 

whence, 


which  is  an  expression  for  the  intensity  of  pressure  at  the 
point  6,  on  the  joint  a  b.  To  solve  for  the  value  of  p', 
the  intensity  of  the  pressure  at  the  point  a,  in  a  similar 
manner  we  may  take  moments  about  a  point  JZ  from  b, 
whence, 


or, 

2W 


When  p'  becomes  zero,  the  trapezoid  reduces  to  a  tri- 
angle as  shown  in  Fig.  3,  with  its  center  of  gravity  at  a 


10 


HIGH    MASONRY    DAM    DESIGN 


distance  from  b  equal  to  J/,  and,  since  the  center  of  pres- 
sure of  W  must  lie  vertically  above  the  center  of  gravity 
of  the  triangle  graphically  representing  it,  we  shall  have, 
pf  =  O,  u  =  J/,  and  Eq.  (9)  reducing  to 


whence, 


2\Y 


(13) 


(14) 


That  is  to  say,  the  maximum  pressure  p  is  twice  the  value 
as  obtained  from  Eq.  (6). 


$  1 

ty/////////////////////////////^^ 


FIG.  3. 


FIG.  4- 


In  Fig.  4  is  represented  a  case  in  which  tension  exists 
over  a  portion  of  the  joint,  p'  is  here  negative. 

Although  both  masonry  and  the  best  hydraulic  cement 
mortar  have  considerable  tensile  strength,  running  up  to 
several  hundred  pounds  per  square  inch  in  tests,  the 
latter,  together  with  the  continued  adhesion  of  the  mortar 
to  the  aggregate  in  concrete,  when  used,  is  of  uncertain 
value  in  this  connection.  The  tensile  strength  is  therefore 
always  neglected  in  considering  the  stability  of  masonry 
dams  or  other  similar  structures,  and  is  an  omission  which 


HIGH    MASONRY    DAM    DESIGN  11 

is  the  more  justifiable  since  it  leads  to  an  error  on  the 
side  of  safety. 

In  the  case  represented  by  Fig.  4,  the  triangle,  whose 
base  is  3^,  and  altitude  p,  is  therefore  alone  considered, 
and  by  taking  moments  about  b,  there  will  result, 


-u (15) 


whence, 

2W 


If  it  is  desirable  to  know  what  the  tension  in  the  joint 
is,  it  may  be  determined  from  Eq.  (12).  As  ^y-<i.o,  the 

resulting  value  is  negative,  hence  denoting  a  tension 
by  that  equation. 

The  pressures  at  a  and  b  may  also  be  determined  as 
follows:  Decomposing  the  resultant  acting  on  any  joint 
into  its  vertical  and  horizontal  components,  V  will  repre- 
sent the  total  normal  or  vertical  pressure,  equal  to  W, 
the  weight  of  masonry  above  the  joint,  plus  the  vertical 
component  of  the  thrust  from  the  water.  The  horizontal 
component  of  the  resultant  is  disregarded,  as  its  effect 
upon  the  joint  is  more  or  less  indeterminate,  and  since 
too,  it  is  assumed  to  be  neutralized  by  the  friction  acting 
in  the  joint. 

The  vertical  component  V,  acting  through  the  point  of 
application  of  the  resultant  R  in  the  joint,  is  therefore 
the  factor  producing  the  difference  in  pressure  between 
a  and  b,  or  the  uniformly  varying  stress. 


12  HIGH    MASONRY    DAM    DESIGN 

Assume  that  at  the  center  of  the  joint,  which  is  not 
necessarily  vertically  below  the  center  of  gravity  of  the 
mass  above,  two  forces  equal  and  opposite  to  each  other, 
and  of  the  same  value  V,  are  applied  normal  to  the  joint. 
The  effect  of  each  is  to  neutralize  the  other,  but  if  we 
consider,  apart  from  the  other  forces,  the  one  acting  down- 
ward, since  it  is  applied  at  the  center  of  figure  it  will 

V 

produce  a  uniform  stress  p  over  the  joint  equal  to  j. 

The  two  remaining  and  equal  forces  V  and  V,  one 
acting  downward  at  the  point  of  application  of  R,  and  the 
other  upward  at  the  center,  form  a  couple  whose  lever 
arm  is  v>  and  the  moment  of  which  is  therefore  Vxv. 
This  moment  produces  a  'uniformly  varying  stress  over 
the  joint,  increasing  the  intensity  at  b  and  decreasing  it 
at  a  by  an  equal  amount. 

To  determine  its  value  we  have  but  to  consider  the  fol- 
lowing: 

M  =  Vv  .......     (17) 

the   moment   caused   by   the   couple   and   producing   the 
varying  stress.     Also, 


where  k  is  the  intensity  of  stress  at  the  maximum  dis- 
tance from  the  neutral  axis;  /,  the  moment  of  inertia 
of  the  section  about  such  an  axis;  and  d\  the  normal  dis- 
tance from  the  neutral  axis  to  that  point  where  k  exists. 
Since  the  neutral  axis  passes  through  the  center  of 
figure  of  the  joint,  the  value  of  di  is  half  the  length  of  the 


HIGH    MASONRY    DAM    DESIGN  13 

joint,  while  /,  the  moment  of  inertia,  equals  Ty3,  if  we 
consider  a  horizontal  section  in  the  plane  of  the  joint  a  b 
extending  back  from  the  plane  of  the  paper  one  unit's 
distance.  Hence, 

M  =  y      kl     £TV3     kl2 

or, 

6Vv 

k==^2- (20) 

Here  k  represents  the  stress  that  must  be  added  to  the 

V 
uniform  stress  j  to  get  the  intensity  of  pressure  at  the 

toe  b  and  the  amount  which  must  be  subtracted  from 

V 

-j-  to  get  the  intensity  at  the  heel  a.     It  is  expressed  in 

pounds  per  square  inch,  but  if  the  distances  are  measured 
in  feet  and  the  forces  in  pounds,  k  will  be  designated  in 
pounds  per  square  foot. 

While  it  is  customary  to  consider  only  the  normal 
component  of  the  resultant  pressure  acting  in  a  hori- 
zontal joint  and  to  assume  it  to  vary  uniformly,  this  is 
probably  correct  only  for  horizontal  joints  in  rectangular 
walls  vertically  loaded  and  not  subjected  to  lateral  pres- 
sures. It  will  be  shown  later  that  the  maximum  stresses 
exist  at  or  near  the  downstream  face,  and  act  in  direction 
parallel  to  and  on  planes  normal  to  that  face.  The  fact 
also  that  acute  edges  do  not  crack  off  in  the  inclined  faces 
of  dams  is  in  itself  a  partial  confirmation  of  the  statement. 

Under  these  circumstances  then,  the  maximum  normal 
pressure  in  a  horizontal  joint  must  be  much  less  than  the 


14  HIGH    MASONRY    DAM    DESIGN 

actual  maximum  pressure  in  the  dam,   and  it  has  been 
assumed  to  bear  the  ratio  to  the  latter  of  about  9  to  13. 

DEVELOPMENT  OF  FORMULAE  FOR  DESIGN. 

Six  series  of  formulae,  designated  by  the  letters  A,  B, 
C,  D,  E,  and  F,  will  now  be  presented,  in  each  of  which 
a  given  set  of  conditions  with  respect  to  the  external 
forces  will  be  involved;  but  as  the  method  of  procedure 
is  practically  the  same  for  all  cases,  only  series  A  will 
be  developed  here. 

The  following  nomenclature  will  be  employed: 

L  =  the  width  of  the  top  of  the  dam  cross-section ; 
/  =  length   of   a  horizontal  joint   of  masonry,    to 

be  determined; 
IQ  =  known  length  of  the  joint  next  above  joint  of 

length  /; 
h  =  depth  of  a  course  of  masonry  (vertical  distance 

between  10  and  /) ; 
P  =  line  of  pressure,  reservoir  full; 
P'=line  of  pressure,  reservoir  empty; 
u  =  distance  from  front  edge  of  the  joint  /  to  the 

point  of  intersection  of  P  with  the  joint  /, 

measured  parallel  to  joint  /; 
y  =  distance  from  back  edge  of  the  joint  /  to  the 

point  of  intersection  of  P'  with  the  joint  /. 

measured  parallel  to  joint  /; 
y0  =  distance   from    back   edge  of   the  joint  1Q  to 

the   point   of   intersection    of   Pr   with    the 

joint  /o,  measured  parallel  to  joint  /0; 


HIGH    MASONRY    DAM    DESIGN  15 

v  =  distance   between   P   and   P'   at   the  joint   I, 

measured  parallel  to  joint  /; 
Y  =  weight   in  pounds   of  a  cubic  foot  of  water 

(62.5); 
7-'  =  weight   in   pounds   of   a   cubic   foot   of   mud 

(75-9°); 

A  =  ratio  of  unit  weight  of  masonry  to  unit  weight 
of  water   (often  assumed  as  J) ; 

Af  =  weight  in  pounds  of  a  cubic  foot  of  masonry; 

H  =  head  of  water  on  joint  /  (vertical  distance  of 
joint  /  below  water  surface); 

H'  =  depth  of  earth  back  fill  over  joint  /  on  front ; 
HI  =head  of  water  on  joint  I  when  ice  acts  at  sur- 
face of  water; 

Hi=rise  of  water  level,  due  to  flood,  wave,  etc., 
above  normal  level  for  full  reservoir; 

hi  =  head  of  water  above  mud  level   (liquid  mud 
of  weight  f) ; 

/*2  =  head  of  liquid  mud  on  joint  /,  on  back; 
a  =  vertical  distance  from  the  top  of  the  dam  to 
the  surface  of  water  (flood) ; 

ai=  vertical  distance  from  the  top  of  the  dam  to 
the  surface  of  water  when  ice  is  considered 
(ai  generally  exceeds  a) ; 
6  =  vertical   distance   from   water  surface   to   top 

of  dam  when  dam  is  overtopped; 
c-=  ratio    of    upward    thrust    intensity,    due    to 
hydrostatic  head  H  (or  HI,  or  hi+h2),  as- 
sumed  to   act   at   heel   of  joint   /    (usually 
assumed  as  J) ; 


16  HIGH    MASONRY    DAM    DESIGN 

Tr= horizontal  ice  thrust  at  water  surface  in  pounds 

(47,000); 

(The  value  here  given,  for  example,  was  used 
in  studies  for  design.  Our  present  lack 
of  exact  data  in  regard  to  ice  pressures 
prevents  more  than  a  speculation  from 
being  made  as  to  a  definite  value  to  be 
assigned  in  any  case); 
=  horizontal  dynamic  thrust  of  water  in  pounds; 

r  =  thrust  of  earth  back  fill  in  pounds  (on  front); 
v?  =  vertical   pressure   on   inclined   upstream   face 
above  joint  /,  in  pounds; 

Q  =  total    area    of    cross-section    of    dam    above 

joint  /o; 

A  =  total    area    of    cross-section    of    dam    above 
joint  /. 

t  =  batter  of  upstream  face  for  vertical  distance  h ; 

s  =  distance  of  line  of  action  of  Wvf  from  upstream 
edge  of  joint  /,  measured  parallel  to  joint  /; 

§  =  angle  that  Ef  makes  with  horizontal; 

a  =  angle  of  slope  of  downstream  face  of  dam 
with  horizontal; 

f}  =  angle  R  makes  with  the  vertical; 

p  =  maximum  allowable  pressure  intensity  at  toe 
(in  pounds  per  square  foot) ; 

q  =  maximum  allowable  pressure  intensity  at  heel 
(in  pounds  per  square  foot)  (p  is  assumed 
less  than  q)  p  and  q  may  be  used  to  signify 
the  calculated,  existent  pressure  intensities 
corresponding  to  P  and  P'  respectively,  for 
the  joint  /. 


HIGH    MASONRY    DAM    DESIGN  17 

/  =  the    coefficient    of    friction    for    masonry    on 

masonry  (usually  0.6  to  0.75); 
5  =  the   shearing   resistance   of   the  masonry   per 

square  unit; 

rH2 

L — =the  horizontal  static  thrust  of  the  water  in 

2 

pounds ; 

rH3 

^-T— =the   moment   of   F  about   any   point   in   the 

joint  /; 

total    weight,    in    pounds,    of    masonry 
resting  on  the  joint  /; 

total    weight,    in    pounds,    of    masonry 
resting  on  the  joint  10; 
resultant  of  F  and  W\ 
R'  =  the  resultant  of  the  reactions ; 
cHlr 


2 


=  upward  thrust  of  water  on  base  /. 


In  the  figures,  hydrostatic  pressures  are  indicated  by 
triangular  and  trapezoidal  areas  included  within  dotted 
lines,  while  ice  pressure  is  shown  to  contrast  HI  with  H. 

As  before,  a  unit  length  of  one  foot  of  dam  will  be 
considered.  Then  the  letters  T,  D,  E,  WV)  A,  A0,  and  H2 
will  signify  volumes. 

It  will  be  observed  that,  where  possible,  the  several 
equations  will  have  been  cleared  of  the  term  4?  thereby 
simplifying  actual  calculations. 

In  the  above  table  c,  in  a  manner,  may  be  considered 
to  provide  for  an  assumption  of  a  certain  proportion  of  the 
joint's  area  being  subjected  to  upward  water  pressure; 
and  the  distribution,  as  evidenced  by  cHlf/2,  varying  from 


18  HIGH    MASONRY    DAM    DESIGN 

a  maximum  intensity  at  the  heel  to  zero  intensity  at  the 
toe,  is  assumed  in  view  of  the  facts  that  the  tendency 
to  open  the  joint  would  begin  at  the  heel  and  a  zero  intensity 
of  upward  pressure  at  the  toe  would  presuppose  an  opening 
with  consequent  flow  at  that  point.  As  the  dam  would 
then  be  failing  in  its  chief  function,  i.e.,  to  retain  water, 
this  flow  is  not  considered  to  exceed  a  slight  seepage. 

In  general  four  ways  are  recognized  in  which  a  masonry 
dam  may  fail : 

1.  By  overturning  about  the  edge  of  any  joint,   due 
to  the  line  of  action  of  the  resultant  passing  beyond  the 
limits  of  stability. 

2.  By    the    crushing    of    the    masonry    or    foundation 
because  of  excessive  pressure. 

3.  By  the  shearing  or  sliding  on  the  foundation  or  any 
joint,  due  to  the  horizontal  thrust  exceeding  the  shearing 
and  frictional  stability  of  the  material. 

4.  By  the  rupture  of  any  joint  due  to  tension  in  it. 
An  unsatisfactory  foundation  might  also  be  mentioned 

as  possibly  leading  to  failure,  and  in  view  of  this,  the 
footing  upon  which  the  dam  rests  should  always  be  most 
carefully  scrutinized. 

To  preclude  failure  from  any  of  the  above  mentioned 
causes,  it  is  the  practice  to  design  the  cross-section  of 
the  dam  with  the  following  conditions  imposed : 

1.  The   lines   of   pressure,    both   for   the   reservoir  full 
and  empty,  must  not  pass  outside  the  middle  third  of  any 
horizontal  joint. 

2.  The    maximum    normal    working    pressure    on    any 
horizontal   joint    must   never    exceed    certain    prescribed 
limits,  either  in  the  masonry  itself  or  in  the  foundation. 


HIGH    MASONRY    DAM    DESIGN  19 

3.  The  coefficient  of  friction  in  any  plane  horizontal 
joint,  or  between  the  dam  and  its  foundation,  must  be 
less  than  the  tangent  of  the  angle  which  the  resultant 
makes  with  a  vertical. 

As  may  be  seen  by  referring  to  the  figures  showing 
the  distribution  of  pressure  on  a  joint,  when  the  resultant 
lies  within  the  middle  third,  tension  can  exist  in  no  part 
of  it,  nor  can  the  safety  factor  be  less  than  two,  if  we 
neglect  to  consider  the  upward  pressure  of  water  perco- 
lating through  any  of  the  joints  or  beneath  the  dam. 


U y . -u >K tc 


To  illustrate  the  conditions  that  obtain  and  to  derive 
the  value  of  the  safety  factor  when  the  resultant  cuts  the 
joint  at  the  extremity  of  the  middle  third,  we  may  take 
the  case  as  shown  in  Fig.  5.  Resolving  R  into  its  horizontal 
and  vertical  components,  and  taking  moments  about  the 
center  of  resistance  e,  the  following  equation  is  obtained: 

F-=W-, (21) 

O       .  O 

where  F  is  the  horizontal  component  of  the  thrust  from 
the  water  behind  the  dam,  acting  at  a  point  %H  above 
the  plane  of  the  joint,  while  W  is  the  vertical  component 
of  the  resultant,  and  as  such,  includes  not  only  the  weight 
of  the  masonry,  but  the  vertical  component  of  the  thrust 


20  HIGH    MASONRY    DAM    DESIGN 

from  the  water  as  well,  provided  the  latter  is  considered 
as  acting  normal  to  the  back  of  the  dam. 

For  the  dam  to  be  on  the  point  of  rotating  about  b, 
the  downstream  edge  of  the  joint,  it  is  obvious  that  the 
resultant  R  must  pass  through  that  point.  Under  these 

IT 

circumstances,  since  the  lever  arm  of  F  is  still  — ,  and  the 

o 

lever  arm  of  W  has  been  increased  to  twice  its  former 
value  or  2  f  — j ,  for  the  above  equation  to  still  hold,  F  must 

also  be  increased  to  twice  its  former  value.  This  would 
indicate  that  when  R  acts  through  the  point  e,  the  value 
of  H  is  only  one-half  as  great  as  is  necessary  to  produce 
overturning;  or,  in  other  words,  that  the  factor  of  safety 

is  two  as  indicated  by  the  ratio  of  — y — .     It  should  be 

observed  however,  that  the  material  near  the  edge  of  the 
joint  will  crush  some  time  before  the  resultant  has  reached 
it,  and  that  therefore  the  factor  of  safety  against  overturning 
with  R  at  the  limit  of  the  middle  third  is  something  less 
than  two. 

When,  however,  the  upward  pressure  of  water  acting 
over  the  joint  due  to  percolation  is  taken  into  considera- 
tion, the  factor  of  safety  will  be  somewhat  modified,  as  the 
following  demonstration  will  make  clear. 

It  is  evident  that  the  forces  acting  upon  the  joint  are, 

rH2 
neglecting  the  reaction,  W,  the  weight  of  masonry,  - — 

the  horizontal  thrust  of  the  water  normal  to  the  back, 
and  — — ,  the  uplift.  Those  tending  to  produce  rotation 


HIGH    MASONRY    DAM    DESIGN  21 

rH2  cHrl 

about  the  downstream  edge   b  are  -  —  and  —  —  ,  the  re- 

2  2 

sultant  of  which  may  be  represented  by  nm,  and  whose 
normal  distance  from  b,  the  center  of  rotation  is  r:  while 
the  force  resisting  this  is  W,  with  a  lever  arm  of  (u  +  V)  . 

If  nm  represent  the  resultant  of  the  overturning  forces 
in  direction  and  magnitude,  and  nd  represent  W,  the  final 
resultant  will  be  found  by  combining  the  two  and  it  will 
act  through  the  point  e,  the  center  of  resistance. 

The  overturning  moment  may  be  written, 


and  the  resisting  moment  by 

=  W  (u  +  V). 


As  the  factor  of  safety  is  the  ratio  of  the  resisting 
to  the  overturning  moment,  it  will  be  represented  by, 

Mo         Mo 


or,  if  the  ratio  of  the  resultant  moment  of  the  vertical 
components,  to  the  resultant  moment  of  the  horizontal 
components  be  considered, 

M0-M2 
Mi 

This  latter  implies  that  the  horizontal  thrust  alone  is 
instrumental  in  the  case  of  overturning  and  that  the  effect 
of  uplift  is  merely  to  reduce  the  resisting  moment. 


22  HIGH    MASONRY    DAM    DESIGN 

It  is  apparent  that  these  two  expressions  will  be  equal 
when  M2  =  0,  i.e.,  when  uplift  is  neglected,  and  also, 
when  the  factor  of  safety  becomes  unity;  i.e.,  at  the  point 
of  overturning.  To  secure  this  by  means  of  the  latter 
consideration  M0  must  equal  Mi  +  M2.  Since  W  is  con- 
stant, either  or  both  of  the  other  factors  may  be  con- 
sidered  to  vary;  but  as  r  has  been  shown  to  be  constant 
also,  both  the  pressure  of  the  water  on  the  back,  and  the 
uplift  must  be  assumed  to  increase  proportionately,  if  the 
resultant  R  is  to  pass  through  the  point  b.  This  seems 
justifiable,  as  the  horizontal  thrust  from  the  water  cannot 
increase  without  a  corresponding  increase  in  the  uplift. 

As  was  stated  previously,  the  frictional  and  shearing 
resistance  of  a  joint  is  assumed  to  withstand  the  tendency 
of  the  horizontal  thrust  to  slide  the  upper  portion  over 
the  lower,  so  that  it  is  quite  customary,  even  though  it 
should  be  investigated,  to  neglect  it. 

For  equilibrium  in  this  regard, 


(22) 


where  F  is  the  horizontal  component  of  the  water's  thrust,  / 
the  coefficient  of  friction,  usually  taken  between  0.6  and 
0.75  for  masonry,  and  5  is  the  shearing  resistance  per  unit 
of  area. 

In  spite  of  the  fact  that  5  has  an  appreciable  value, 
and  particularly  so  for  monolithic  masses  of  "  cyclopean 
masonry,"  the  value  is  practically  indeterminate,  and 
consequently  usually  ignored.  Numerous  attempts  have 
been  made  however,  to  write  expressions  for  it,  the  most 
rational  of  which  depends  upon  the  trapezoidal  law  of 


HIGH    MASONRY    DAM    DESIGN  23 

the  distribution  of  normal  stress;  but  this  too  is  unsatis- 
factory from  a  practical  standpoint.*  We  shall  neglect 
S,  therefore,  in  the  previous  equation,  whence, 


.......     (23) 

which  gives  at  the  limit, 

p 


(24) 


In  every  design  the  imposed  conditions  for  equilibrium 
result  in  a  cross-section  in  which  the  back  has  very  much 
less  of  a  batter  than  the  front.  It  may  be  shown  also,* 
that,  as  the  shear  along  either  face  is  zero,  the  greatest 
intensity  of  stress  will  act  in  a  direction  parallel  to  the 
face  at,  and  near,  the  edge.  Since  the  horizontal  compo- 
nent of  the  pressure  is  ignored,  this  implies  that  the 
greatest  vertical,  or  normal  working  intensity  of  pres- 
sure must  be  less  at  the  downstream  face  where  the 
inclination  is  greater  than  at  the  heel,  in  order  that  the 
components  parallel  to  the  respective  faces  shall  be  ap- 
proximately equal.  This  is  accomplished  by  using  a 
smaller  vertical  normal  working  stress  at  the  toe  than 
at  the  heel. 

As  the  upstream  face  of  a  masonry  dam  is  vertical 
for  a  considerable  distance  from  the  top,  and  then  be- 
comes only  slightly  inclined  to  it,  it  is  customary  to 
consider  the  thrust  from  the  water  as  acting  horizontally. 
This  is  the  more  justifiable  since  the  vertical  component 
of  the  water  resting  upon  the  upstream  face  of  the  dam 

*  See  Appendix. 


24  HIGH    MASONRY    DAM    DESIGN 

causes  an  overturning  moment  about  the  center  of  re- 
sistance, opposite  in  direction  to  that  induced  by  the 
horizontal  thrust,  and  hence  is  an  error  on  the  side  of 
safety. 

It  must  be  evident  from  the  equation  of  pressure, 
p  =  rah,  that  where  only  this  governs  the  resulting  theo- 
retical cross-section,  it  will  be  triangular  in  form  with  the 
apex  at  the  surface  of  the  water;  but  where  it  is  intended 
there  shall  be  no  flow  over  the  crest  of  the  dam,  it  is 
customary  to  carry  the  masonry  some  distance  above 
the  elevation  of  the  water  in  the  reservoir,  not  only  to 
allow  for  fluctuations,  but  because  of  economic  condi- 
tions or  to  provide  for  a  foot  or  carriage  way.  The  super- 
elevation and  the  width  of  top  are  therefore  arbitrarily 
assumed  and  should  be  taken  at  about  TV  the  height 
of  the  dam,  with  a  minimum  width  of  5  feet  and  a  maxi- 
mum superelevation  of  20  feet. 

As  no  equation  can  be  written  simultaneously  ex- 
pressing the  three  conditions  of  stability,  i.e.,  that  the 
resultant  lie  within  the  middle  third,  that  the  maximum 
pressures  shall  not  exceed  certain  limits,  and  that  the 
horizontal  components  shall  not  cause  sliding,  it  be- 
comes necessary  to  determine  the  length  of  joints,  usually 
taken  vertically  10  feet  apart  for  a  depth  of  about  100 
feet  and  increasing  to  20  or  30  feet  below,  by  the  aid  of 
that  equation  involving  the  limiting  conditions  known 
to  apply,  in  order  that  the  cross-section  be  a  minimum, 
and  then  to  test  the  joint,  if  necessary,  by  the  other  two. 
Generally  speaking  the  third  condition  will  be  found  to 
hold  if  the  joint  has  been  designed  in  accordance  with  the 
other  two. 


OF   THE 

UNIVERSITY 


HIGH    MASONRY    DAM    DESIGN  25 

Considering  Fig.  5,  in  which  /  is  the  length  of  joint, 
it  is  seen  to  be  divided  into  three  parts,  u,  v,  and  y, 
and  from  this  what  may  be  called  the  fundamental  equa- 
tion of  the  entire  design  can  be  written. 

l=u  +  v  +  y  .......     (25) 

If  M  represents  the  overturning  moment  about  e,  then 
we  have  that  at  the  limit  of  the  middle  third, 

M  =  F-=Wv,      .....     (26) 

O 

or, 

M  . 

v=w  ........  (27) 

As  the  analysis  will  result  in  a  cross-section  polygonal 
in  outline,  composed  of  trapezoids  with  bases  /  and  /o 
and  altitudes  h,  we  may  write  a  general  equation, 


....        (28) 
\    ^     / 

or, 


whence, 

A=Ao+[*-^)h,         .....     (29) 

and  since, 

W      . 

-r=A0  + 


then, 

M 


v  = //  ,  i\    »        (30) 


26  HIGH    MASONRY    DAM    DESIGN 

which  value  of  v,  if  substituted  in  Eq.  (25)  gives, 


M 
Jr 


>    -   -   -    (31) 


The  above  Eq.  (31)  is  a  modification  of  Eq.  (25)  and, 
when  proper  values  have  been  assigned  to  u  and  y,  depend- 
ing upon  the  existing  conditions,  is  used  throughout  the 
entire  design  in  the  determination  of  the  length  of  joints. 

In  the  upper  rectangular  portion  of  the  dam,  where 
there  is  an  excess  of  material  above  that  required  by  the 
static  pressure  of  the  water,  it  will  be  found  unnecessary 
to  consider  failure  from  crushing,  as  the  maximum  normal 
pressures  are  well  below  the  allowed  working  pressure, 
and  consequently  the  depth  at  which  the  section  ceases 
to  be  rectangular  will  be  fixed  by  the  fact  that  the  re- 
sultant may  not  pass  outside  the  middle  third.  The 
algebraic  expression  for  this  condition  is, 

u  =  J  I  for  reservoir  full,        .     .     .     .     (32) 
y  =  i  I  for  reservoir  empty.        .     .     .     (33) 

Below  this  rectangular  portion,  trapezoidal  sections 
will  be  found.  At  the  base  of  the  rectangle,  l  =  l$  =  L, 

u  =  —,   and,   since   the   center   of   gravity   of   the   figure   is 

O 

vertically  above  the  center  of  the  joint,  y  =  —. 

If  we  wish  to  determine  the  depth  to  which  the  rectan- 
gular portion  extends,  we  may  do  so  by  the  use  of  Eq. 


HIGH    MASONRY    DAM    DESIGN  27 

(31),   which,  as   shown,  must  involve   the  condition  that 
the   resultant    shall   just   touch   the   limit   of    the   middle 

third,  i.e.,  u=~.     Substituting  in  Eq.    (31),  u  =  -,  ^  =  ^' 

O  _O 

and  remembering  that  A0  =  o 


_L          6J          L 
= 


whence,  by  dividing  by  L, 

#3  #3 


or,  solving  for  H, 

....     (34) 


If  H=h  then  a=o,  and  Eq.  (34)  reduces  to, 

(35) 

At  this  depth  the  rectangle  ceases,  the  sections  become 
trapezoidal,  the  back  face  is 
still  vertical  but  the  front 
face  /&',  is  inclined  in  order 
to  increase  the  length  of  the 
successive  joints  and  thus 

maintain     the     resultant    for 

i 

the  reservoir  full  at  the  down-  FIG.  6. 

stream  limit    of    the    middle 

third.     For  a  considerable  distance  below  the  rectangular 

section  therefore,  Eq.  (31)  will  be  used  with  u  =  —  to  de- 

o 


28  HIGH    MASONRY    DAM    DESIGN 

termine  the  length  of  joint,  and  the  back  face  will  remain 
vertical,  but  for  each  new  joint  the  resultant  for  the  reser- 
voir empty  will  approach  nearer  and  nearer  to  the  limits 

of  the  middle  third,  until  finally  y=—. 

o 

It  is  therefore  expedient  to  determine  the  value  of  y 
under  these  conditions  to  learn  exactly  at  what  vertical 

depth  or  joint  this  value  of  y  first  equals  -. 

o 

To  do  this,  moments  are  taken  about  the  vertical 
face,  for  both  AQ  and  the  trapezoid,  the  latter  being  found 
by  dividing  the  trapezoid  into  a  rectangle  and  a  triangle; 
its  value  is, 


and  hence, 


Substituting  the  value  of  A  from  Eq.  (29)  in  the  above, 
and  solving  for  y  there  results, 


(36) 


This  gives  a  value  of  y  to  be  substituted  in  Eq.   (31) 
while   the   value   of   w  =   /,    which   has    been    maintained 


HIGH    MASONRY    DAM    DESIGN  29 

since  leaving  the   bottom   of   the   rectangular  section,   is 
substituted  also.     There  then  results  by  reduction, 


>  •  •  (37) 


.vhich  is  the  equation  used  in  the  determination  of  the 
length  of  joint  from  the  foot  of  the  rectangular  section 
down  to  that  joint  where  Eq.  (36)  first  gives  a  value  of 

y  =  -.     At  this  point  the  back  face  must  be  made  to  slope, 
o 

while  u=y  =  \l  is  substituted  in  Eq.  (31)  to  obtain  the 
following  : 


w> 


which  will  determine  the  length  of  the  joints. 

The  second  condition  will  be  a  factor  from  here  on, 
for  below  this  section  at  some  point,  the  intensities  of  the 
pressures  at  the  toe  will  gradually  approach  and  finally 
equal  the  allowable  limit  p,  and  the  length  of  the  joint 
will  depend  primarily  upon  this.  It  is  therefore  necessary, 
after  each  application  of  Eq.  (38)  to  see  if  the  limiting 
pressure  p  at  the  toe,  which  is  smaller  than  q,  at  the  heel, 
has  been  reached.  Its  value  is  derived  from  the  equation 

p  =  —r  = — j- >  and  when  the  limiting  value  of  p  has  been 
realized  the  value  of  u  thereafter  must  be  derived  from, 

2/  £/2 


. 

(39) 


30  HIGH    MASONRY    DAM    DESIGN 

in  which  u,  is  seen  to  be  dependent  upon  the  normal  working 
pressure  p  at  the  toe.  (Eq.  (39)  follows  directly  from 
Eq.  (9)). 

There  is  some  distance  below  this  joint,  however, 
where  y  still  remains  equal  to  J/,  while  the  value  of  u 
is  being  determined  from  the  above  Eq.  (39).  Under 
these  circumstances,  /  will  be  found  from  the  following 
after  substituting  the  values  of  y  =  $l,  u  from  Eq.  (39) 
and  A  from  Eq.  (29),  all  in  Eq.  (31). 


(40) 


This  equation  will  be  used  until  a  joint  has  been  reached 

2.4  Jr 
where  the  application  of  q  =  — j —  shows  its  value  to  be 

equal  to  or  greater  than  that  prescribed  for  q.     Here  y 
will  be  determined  by 

2l        ql2 


in  which  it  is  seen  to  depend  on  q. 

When   this   point   has   been   reached,   u  will   take   its 

value    from    u=—  -   ^      ,    and    y    from    the     equation 

2l        ql2 
y  =  --  t  which  must   be  substituted   in  Eq.  (31)  to 

determine  /.     This  will  give  after  reduction, 


H* 

'11         +-      •  •  •  (42) 


HIGH    MASONRY    DAM    DESIGN 


31 


All  joints  below  this  point  will  be  found  by  this  last 
equation. 

Summing  up,  we  may  say  that  Eqs.  (34),  (37),  (38),  (40), 
and  (42),  are  the  five  equations  to  be  used  in  determining 
the  length  of  joints  from  the  top  down.  Strictly  speaking 
Eq.  (34)  gives  the  depth  at  which  the  rectangular  portion 
ceases,  while  Eq.  (37)  gives  the  length  of  joints  from  the 
base  of  the  rectangle  down  to  where  y  =  \l\  Eq.  (38)  the 
length  of  joints  from  the  point  where  y  =  %l  to  where  p 
reaches  its  limiting  value;  Eq.  (40)  the  length  of  joints 
from  the  point  where  p  equals  its  limiting  value  to  where 
q  equals  its  limiting  value  and  Eq.  (42)  gives  the  length 
of  all  joints  below. 

Eqs.  (34)  and  (37)  involve  the  value  of  y,  which  is 
obtained  with  respect  to  the  vertical  back,  but  when 
that  face  begins  to  slope  it  is  necessary  to  determine  it 
with  regard  to  the  back  edge  of  the  joint  in  question. 


I  w0 


FIG.  7. 

In  Fig.  7,  mn  represents  the  back  face  of  the  dam  and 
/  is  the  batter  to  be  determined  by  taking  static  moments  of 
A  and  A0  about  the  back  edge,  m,  of  the  joint. 

The  trapezoid  of  the  figure  is  composed  of  the  triangles 
ht/2  and  (l-l0-t)h/2  and  the  rectangle  hl0. 


32  HIGH    MASONRY    DAM    DESIGN 

By  taking  moments  about  the  edge  m, 


.  (43) 


For  Eqs.  (38)  and  (40)  the  value  of  y  must  be,  as  be- 
fore, taken  equal  to  J/,  while  A  has  the  usual  value  of 
Ao+(l  +  l0)h/2.  Substituting  these  in  Eq.  (43)  and  re- 
ducing: 

(44) 


For  the  joints  to  which  Eq.  (42)  applies  the  value  of  y  is 
to  be  taken  from  Eq.  (41)  as  was  done  before.  In  this 
case: 


By  substituting  this  value  in  the  first  member  of  Eq.  (43) 
and  reducing: 


t=  '  '     '     (45> 


After  the  value  of  /  is  found  by  the  use  of  Eqs.  (38), 
(40)  or  (42),  t  can  at  once  be  determined  for  the  same 
joint  by  either  Eq.  (44)  or  Eq.  (45). 

In  this  manner  an  entire  theoretical  cross-section  can 
be  determined.  It  will  be  noticed  that  the  location  of  the 
center  of  pressure  in  the  middle  third  of  the  joint  is  the 
governing  condition  in  the  upper  part  of  the  dam,  while 
the  lower  portion  is  fixed  by  the  limiting  pressures  p  and  q. 


HIGH    MASONRY    DAM    DESIGN  33 

The  difficulties  preventing  the  forming  of  a  simple  working 
equation  for  the  entire  cross-section  arise  from  the  fact 
that  the  governing  conditions  are  not  introduced  simul- 
taneously nor  in  the  same  joint. 

By  taking  h  of  the  proper  value,  a  polygonal  cross- 
section  may  be  determined  by  the  preceding  formulae. 
This  cross-section  can  be  then  modified  by  drawing  what 
may  be  called  "mean"  lines,  straight,  broken  or  curved 
along  the  theoretical  faces  so  as  to  adapt  the  latter  to  a 
practical  arrangement  and  treatment  of  the  joints  and 
facing  blocks,  which  may  be  of  cut  stone  or  concrete. 

The  conditions  which  have  governed  the  analysis  are 
essentially  those  of  Rankine,  i.e.,  the  center  of  pressure 
has  in  all  cases  been  kept  within  the  middle  third  of  the 
joint  and  the  greatest  intensity  of  pressure,  either  at  the 
front  face  or  back,  has  not  been  allowed  to  exceed  the 
limit  p  or  q. 

Series  A,B,C,  D,  E,  and  F,  and  For  mules  for  Investigation. 

As  noted  earlier  six  separate  series  of  formulae  for 
investigation  have  been  derived  and  they  will  be  here  set 
forth  in  suitable  form  for  easy  reference  and  use.  As  they 
have  been  developed  by  the  method  just  outlined  it  is 
unnecessary  to  follow  out  the  derivation  of  each  series, 
although  there  exist  some  detailed  differences  in  the  treat- 
ment of  each.  These  details  however,  would  become 
evident  to  anyone  following  the  deductions  throughout. 

Various  conditions  of  "  loading,"  with  approved  as- 
sumptions, such  as  pressure  due  to  expanding  ice  at  the 
water  surface,  upward  water  pressure  on  the  base,  etc., 


OF  THE 

UNIVERSITY 


34  HIGH    MASONRY    DAM    DESIGN 

referred  to  in  the  table  of  nomenclature  previously  given, 
have  been  introduced  and  are  specifically  stated  for  each 
case. 

It  will  be  recalled  that  Eq.  (31)  is  the  fundamental 
expression  for  finding  the  length  /,  of  any  joint;  and,  as 
the  several  conditions  are  introduced,  that  the  "  M  "  must 
in  each  case  signify  the  total  overturning  moment  and 
not  merely  the  moment  of  the  static  water  pressure  on 
the  back. 

The  development  of  a  cross-section,  by  any  one  of  the 
following  series,  may  comprise  five  stages,  each  stage 
representing  the  introduction  of  a  governing  condition. 
Hence,  for  each  stage  there  obtains  a  main  equation  for 
finding  the  length  of  joint  /,  each  main  equation  being 
supplemented  by  secondary  equations  for  y,  M,  and  /; 
p  and  q. 

It  may  be  necessary  to  employ  more  than  one  of  the 
series  of  equations  in  determining  a  cross-section. 

For  ready  reference,  the  five  stages  will  be  set  forth 
and  depicted  in  order  as  follows: 

Stage  I. — This  stage,  it  will  be  remembered,  extends 
from  the  top  of  the  dam  to  the  joint  where  the  front  face 
commences  to  batter.  It  is  the.  rectangular  section. 
y>\L\u^\L  (see  Fig.  8).  (Ice  pressure  is  purposely 
omitted  in  Fig.  8  to  prevent  confusion  of  letters  in  small 
space.) 

Stage  II. — This  stage  extends  from  the  lower  limit  of 
Stage  I  to  the  point  where  the  back  face  commences  to 
batter.  «  =  }/;  y2.$l  (see  Fig.  9). 


HIGH    MASONRY    DAM    DESIGN 


35 


Stage  III. — This  stage  extends  from  the  lower  limit  of 
Stage  II  to  the  point  where  the  intensity  of  pressure  on 


Flood 


FIG.  8. 


the   toe   has   reached   the   maximum   allowable   intensity 
In  this  stage  u  =  \l\y  =  \l  (see  Fig.  10). 


Flood 


W-AAy 


FIG.  o. 


Stage  IV. — This  stage  extends  from  the  lower  limit  of 
Stage  III  to  the  point  where  the  pressure  intensity  on 


36 


HIGH    MASONRY   DAM    DESIGN 


the  heel  has  reached  the  maximum  allowable  intensity. 
For  this  stage  u>\l\  y  =  %l  (see  Fig.  10). 

Stage  V. — In    this    stage    the    limiting    intensities    of 
pressure  at  both  toe  and  heel  having  been  reached,  y>%l\ 


FIG.  10. 


This  stage  extends  from  the  lower  limit  of  Stage  IV 
downward.  (See  Fig.  10.) 

The  following  secondary  formulae,  supplementary  to 
the  main  equations  of  all  series,  with  substitutions  as  noted, 
are  arranged  in  order  corresponding  with  the  preceding. 


HIGH    MASONRY,  DAM    DESIGN 


37 


Stage  I. 


Stage  III. 
u=\l 

-v 


With  the  condition  of  hydrostatic  up- 
ward pressure  on  the  base  obtaining,sub- 
stitute  the  formulae  in  this  column  in 
place  of  those  corresponding,  as  indicated- 


*  A  A 


2  J A 


38  HIGH    MASONRY    DAM    DESIGN 

Stage  IV. 


3^\  (limiting 
/     \2      I  )  intensity) 


Stage  V. 

pi* 


4   


/    _3^\  (limiting 
\        I  /intensity) 


2AlA(2  _  $y\  (limiting 
I     \        I  )  intensity) 


pl* 


If  7  enter  the  following  tormulse,  H  above  becomes 
H\.  (See  Figs.  9  and  10). 

The  first  column  of  formulas  just  given  would  apply, 
with  the  condition  of  upward  pressure  on  the  base  due 
to  hydrostatic  head,  if  a  proper  value  of  u  corresponding,  be 


HIGH    MASONRY    DAM    DESIGN  39 

taken,  that  is  if  the -excursion  of  the  force  A  A?,  resulting 
from  the  effect  of  all  other  forces  on  AAf  be  considered, 
rather  than  the  effect  of  all  the  other  forces  on  the  re- 
sultant vertical  force. 

It  will  be  found  expeditious  to  design  a  section,  where 
ice  pressure  at  the  level  of  full  reservoir  is  to  be  considered 
in  connection  with  the  water  surface  at  some  higher  flood 
level,  first  by  series  of  formulae  containing  T  (cf .  Series  B  and 
D)  and  then  to  investigate  successive  bases,  or  joints,  thus 
obtained  (beginning,  for  a  high  masonry  dam,  usually 
at  a  base,  or  joint,  about  100  feet  from  the  top  of  the  dam) 
with  series  of  formulae  lacking  T,  or  the  ice  pressure  con- 
dition (cf.  Series  A  and  C).  A  base  will  ultimately  be 
obtained  by  these  supplementary  "  Flood  level  "  calcula- 
tions greater  than  the  base  at  its  same  elevation  as  pre- 
viously determined  by  the  "  Ice  Pressure  "  design. 

Continuing  with  the  design  by  means  of  the  "  Flood  level  " 
formulas  to  the  bottom  of  maximum  height  required  will 
determine  the  minimum  cross-section  area  to  meet  the 
conditions  both  of  "  Flood  "  and  of'"  Ice."  It  should  be 
remarked  in  this  connection  that  when  a  reservoir  level 
is  rising  due  to  flood  conditions  prevailing,  it  is  evident 
that  ice  formation  cannot  develop,  or,  in  other  words,  the 
two  conditions  cannot  be  coexistent,  hence  the  difference 
in  designation  of  hydrostatic  heads  corresponding.  (See 
Figs.  9  and  10.) 


40  HIGH    MASONRY    DAM    DESIGN 

SERIES  A. 

Conditions:     Overturning    moment    due    to    horizontal 
static  water  pressure  on  back  of  dam  only. 

Stage  L 
Stage  II. 


Stage  III. 


Stage  IV. 

P 
Stage  V. 


SERIES  B. 

Conditions:   Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  and 

(b)  Ice  pressure  applied  at  distance  (a\)  from  top. 


Stage  I. 


HIGH    MASONRY    DAM    DESIGN  41 

Stage  II. 


Stage  III. 


Stage  IV. 


Stage  V. 


SERIES  C. 

Conditions:    Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back. 

(b)  Upward  water  pressure  on  base.     Pressure  intensity 
decreasing  uniformly  from  cH?  at  heel  to  zero  intensity 
at  toe. 

Stage  I. 
H  = 

Stage  II. 


Stage  III. 


42  HIGH    MASONRY    DAM    DESIGN 

Stage  IV. 


— ( ~r^  +  A) ) ,  which  reduces  to 
p  \  h 


Stage  V. 


—  T-J  (  ~r^  +  ^o  )  ,  which  reduces  to 


SERIES  D. 

Conditions:    Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  (head  =  HI). 

(b)  Ice  pressure  applied  at  distance  (ai)  from  top. 

(c)  Upward  water  pressure  on  base.     Pressure  decreasing 
uniformly  from  cHif  at  heel  to  zero  .in  tensity  at  toe. 

Stage  I. 
Hl= 

Stage  II. 


HIGH    MASONRY    DAM    DESIGN  43 

Stage  III. 


Stage  IV. 


which  reduces  to 


Stage  V. 

cHi 


reduces  to 


In  the  preceding  series  of  equations  it  will  be  observed 
that  the  final  expressions  for  /  in  stages  IV  and  V  are 
very  similar,  and  that  the  quantity  c  in  equations  (a)  of 
these  stages  disappears  in  equations  (6).  Equations  (b) 
of  course,  are  to  be  used  for  purposes  of  calculation  of 
cross-sections. 


44 


HIGH    MASONRY    DAM    DESIGN 


SERIES  E. 

Conditions:     (See   Fig.    n)    Ice   pressure   neglected   in 
Fig.  ii.     Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  (head=/^i). 

(b)  Ice  pressure  applied  at  distance  (a\)  from  top. 

(c)  Upward  water  pressure  on  base;    pressure  intensity 
decreasing  uniformly  from  c(hi+h2)r  at  nee^  to  zero  m~ 
tensity  at  toe. 

(d)  Mud  (liquid)  pressure  on  back  (head  h2) ,  commencing 
at  distance  h2  above  joint  in  question.     Weight  of  mud  =  f . 
As  before,  if  T  be  equated  to  zero,  a\  becomes  equal  to 
a,  in  the  formulas. 


FIG.  ii. 
Stage  I.     (hi,  of  known  value,  h2  to  be  determined.) 


Stage  II. 


~(hi+h2) 


HIGH    MASONRY    DAM    DESIGN  45 

For  trapezoidal  section  at  top,  make  AQ  =  o  and  ^0  =  0 
and  k=L  in  Stage  II.     This  applies  generally. 

Stage  III. 


Stage  IV. 


Stage  V. 


)  (3*1*2 


From  a  study  of  the  formulae  thus  far  developed  it 
will  be  observed  that  by  reducing  certain  conditions  to 
zero,  with  their  corresponding  quantities,  the  main  equa- 
tions of  a  given  series  reduce  to  those  of  a  simpler  series. 

For  instance  — 

In  Series  B  make  T  (for  ice  pressure  condition  of  load- 
ing) equal  to  zero  and  H\=H  and  a\=a  and  the  main 
equations  of  that  series  reduce  to  Series  A  equations. 

In  Series  C,  by  making  c  (for  upward  water  pressure 
condition)  equal  to  zero  in  main  equations  of  Stages  I,  II, 
and  III  and  also  in  equations  (a)  of  Stages  IV  and  V,  the 
equations  of  Series  C  reduce  to  those  of  Series  A. 

Likewise,  by  making  the  proper  eliminations  and  sub- 
stitutions, Series  E  will  reduce  to  Series  D,  C,  B  or  A. 


46 


HIGH    MASONRY    DAM    DESIGN 


SERIES  F. 

This  series  consists  of  general  formulae  for  a  number 
of  imposed  conditions  of  loading.  For  any  given  case, 
the  terms  of  factors  expressing  those  conditions  not  ap- 
pertaining must  be  eliminated  by  equating  them  to  zero. 
(See  Fig.  12.) 


FIG.  12. 


Conditions  for  General  Formula. 

Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  (head=/h). 

(b)  Upward  water  pressure  on  base;    pressure  intensity 
decreasing  uniformly  from  cHr  or  c(hi+h2)r,  at  heel  to 
zero  intensity  at  toe. 

(c)  Mud  (liquid)  pressure  on  back  (head  hz)  as  before. 

(d)  Dynamic  pressure  of  water,  Dr. 

(e)  Water  flowing  over  top   of  dam,   weight  of  water, 
of  depth  6,  on  top  of  dam  being  neglected. 


HIGH    MASONRY    DAM    DESIGN  47 

For  condition  of  water  not  overtopping  dam,  b  =  o  and 
£>  =  o. 

For  condition  of  no  dynamic  pressure,  D  =  o. 

For  condition  of  no  upward  water  pressure,  c  =  o. 

For  condition  of  no  mud  (i.e.,  mud  being  replaced  by 
water)  make  h2  =  o,  h\=H. 

Stage  I. 

Rectangular  cross-section  at  top  or  rectangular  dam, 
1  =  10=L. 

This  may  fall  under  either  of  two  cases,  viz.  — 

Case  (i) 

Condition:  h\=H\  h2  =  o. 


Case  (2) 

Condition:  hi  of  known  value;  h2  to  be  determined. 


As  in  the  preceding  series,  the  value  of  H  or  h2,  of 
Stage  I  may  be  determined  by  several  successive  trial  sub- 
stitutions. 

Stage  II. 

(a)  Trapezoidal  cross-section  at  top  of  dam  or  trape- 


48  HIGH    MASONRY    DAM    DESIGN 

zoidal  dam  (spillway)  front  face  battered.     (A0  =  o,  1Q 
and  yo^=o.)     Note:    For  a  triangular  dam  /o  =  o,  also. 


(b)  Trapezoidal  section  continued  (front  face  battered) 


— 


Stage  III.  —  Both  faces  battered. 


Stage  IV.  —  Limiting  intensity  of  pressure,  p,  introduced 


Stage  V.  —  Limiting  intensities,  p  and  q 


HIGH    MASONRY    DAM    DESIGN  49 

The  increased  number  of  overturning  loads  then,  tend 
to  render  the  right-hand  members  of  the  various  equations 
more  involved;  though  after  a  little  practice  one  may 
easily  carry  through  a  design  with  surprising  rapidity. 
The  slide  rule  may  be  used  to  great  advantage  and  it  is 
suggested  that  the  results  be  tabulated  as  determined, 
in  some  such  form  as  the  following: 


TABLE  OF  RESULTS. 


c=  ,  r=   ,  etc. 


Joint 
No. 

H 

H* 

h 

H  +  a 

Ao 

A 

lo 

lo2 

I 

yo 

y 

14 

t 

P 

Q 

Aoyo 

Etc. 

I 

2 

3 

4 

Etc. 

The  effect  upon  the  calculation  of  a  cross-section,  of 
backfill  on  the  downstream  face  could  of  course  be  cared 
for  by  introducing  that  condition  into  the  preceding  series 
of  equations;  but  as  this  effect  as  computed,  would  be, 
in  any  case,  largely  dependent  upon  assumptions  which 
may  vary  widely  and  as  the  placing  of  backfill  is  generally 
a  later  consideration  with  respect  to  construction,  the 
propriety  of  such  introduction  at  that  stage  of  design  is 
questionable. 

In  the  following  formulae  for  investigation  therefore, 
the  general  conditions  of  an  earth  thrust  acting  at  the 
downstream  face  and  of  a  vertical  component  of  thrust  of 
material  on  the  upstream,  inclined  face  of  the  dam,  are 
introduced. 

By  any  of  these  formulas  the  position  of  the  line  of 
resistance  for  any  given  cross-section  and  respective  con* 


50  HIGN    MASONRY    DAM   DESIGN 

ditions  may  be  determined  with  regard  to  any  horizontal 
joint  and  its  downstream  edge;  the  value  of  u  being  the 
quantity  to  be  sought.  ^ 

Any  condition  may  be  disregarded  by  equating  its 
term  to  zero. 

The  first  expression  below  contains  all  of  the  conditions 
heretofore  considered  with  the  additional  ones  just  stated; 
and  from  it  follow  the  succeeding  expressions  for  u.  It 
should  be  remembered  that  the  term  T  cannot  be  coex- 
istent in  any  expression  for  stability  with  b  and  therefore 
with  D.  Nevertheless  all  of  these  terms  are  written  with 
the  understanding  that  the  proper  eliminations  be  always 
made.  Three  general  group  equations  will  be  written. 

Formulae  for  Investigation. 

First,  Conditions  of  retained  mud,  water,  overtopping, 
etc.  (see  Fig.  12). 


[„  H'sinOJ  +  a)! 

+6E\(!-y)  sm£- 
_,_  3        sin  a      J 

y 


Whence,  for  conditions  of  retained  mud,  water,  etc.,  but 
no  overtopping,  by  making  6  =  0  and  D  =  o,  there  follows 
(see  Fig.  n)  : 


HIGH    MASONRY    DAM    DESIGN  51 

From  this  last  expression  for  u,  for  conditions  of  re- 
tained water,  etc.,  but  neither  mud  nor  overtopping,  by 
making  hi=H\;  h2  =  o,  there  is  obtained: 


-- 

6E\f.       .     .          H'sm(d  +  a 

TT    (l  —  y)  sin  d  ----  r— 
#iLv  3        sma 


As  in  the  equations  for  design,  when  T  =  o,Hi=H.  (See 
Figs.  9  and  10.)  If  H'  is  of  such  depth  that  the  downstream 
batter  of  the  cross-section  varies  considerably,  an  approxi- 
mate solution  is  possible  by  assuming  some  average  batter 
for  the  lower  portion.  The  expression  for  earth  thrust  is 
general  ,  as  is  evidenced  .  After  u  is  determined  for  each  j  oint  , 
the  intensities  of  maxima  pressures  can  be  determined 
for  the  given  cross-section,  the  general  expression  for  p, 
corresponding  to  above  expressions  for  u,  being: 


In  connection  with  the  computation  for  the  value  of 
y  in  an  investigation,  as  indicated  above,  it  is  necessary 
to  obtain  the  position  of  the  centroid  of  a  trapezoid  with 
respect  to  the  back,  or  upstream,  edge  of  the  joint  in 
question.  The  following  expression  for  xt  in  connection 
with  Fig.  13,  may  prove  convenient: 


52 


HIGH   MASONRY    DAM    DESIGN 


In  the  studies  for  design,  referred  to  before,  the  analytic 
work  should  be  checked  throughout  by  the  graphic  method 
wherever  possible.  This  should  always  be  done  both  in 
designing  and  investigating  cross-sections. 

It  should  be  stated  here 
that,  after  a  cross-section  has 
been  fixed  upon  for  a  given 
dam  and  the  faces  drawn  to 
chosen  batters  and  curves,  the 
entire  cross-section  should  be 
investigated  as  just  outlined 
so  as  to  give  the  actual  values 
for  this  final  cross-section. 

Again,  in  comparing  different  cross-sections,  especially 
of  different  dams,  by  superimposing,  their  water  lines 
should  be  made  to  coincide  and  not  their  tops  for  a  fair 
comparison. 


FIG.  13. 


APPENDIX   I 

RECENT  CONSIDERATIONS   OF   THE   CONDITION  OF  STRESS 
IN   A   MASONRY   DAM 

CONSIDERABLE  discussion  has  been  raised  within  the 
past  few  years,  by  criticisms  being  leveled  at  the  present 
general  procedure  in  the  design  of  high  masonry  dams. 
This  has  properly  perhaps,  been  more  pronounced  abroad 
than  in  this  country,  since  the  matter  may  be  said  to  have 
been  precipitated  by  the  publication  of  a  paper  by  Mr. 
L.  W.  Atcherley  of  London  University,  "  On  Some  Dis- 
regarded Points  in  the  Stability  of  Masonry  Dams."* 

It  is  the  purpose  to  outline  the  analysis  as  presented 
there,  and  to  call  attention  to  some  of  the  discussion  which 
followed,  in  order  to  indicate  the  status  of  the  theory 
involved  in  the  design  of  such  structures. 

The  paper  referred  to  takes  exception  to  current 
practice  in  regard  to  the  matter  of  design  and  indicates 
a  need  for  both  revision  and  extension  in  the  analysis, 
and  then,  supplementing  the  generally  accepted  ideas  as 
to  the  distribution  of  normal  stress  on  horizontal  planes, 
by  an  assumption  as  to  the  shear  on  these  planes,  proceeds 
to  show  that  peculiar  and  unexpected  conditions  arise. 

*  Dept.  of  Applied  Mathematics,  University  College,  University  of 
London.     Drapers'  Company  Research  Memoirs.     Technical  Series  II. 

53 


54  HIGH    MASONRY    DAM    DESIGN 

It  is  a  fact  that,  owing  ist,  to  the  manner  in  which 
masonry  structures  are  built,  i.e.,  of  a  mixture  of  stone 
and  cement,  and  2d,  to  the  nature  of  the  sections  at  the 
springings  or  areas  of  support,  it  is  practically  impossible 
to  apply  to  them  the  general  theory  of  elastic  bodies. 
Consequently,  the  treatment  as  it  is  employed  to-day 
has  been  developed,  but  only  by  the  use  of  certain  assump- 
tions which  it  may  be  shown  are  not  precisely  exact. 

The  basis  of  the  present  investigation  rests  upon  the 
four  following  formulae,  in  which  the  usual  distribution 
of  normal  unit  stress  on  horizontal  planes  is  accepted, 
but  to  which  is  added  an  assumed  condition  as  to  the 
distribution  of  horizontal  shear. 


(2) 

,   \ 
(3) 

«> 


c  =  distance  along  the  horizontal  joint  from  the  cen- 

troid  to  the  point  of  application  of  the  resultant. 

d  =  distance  along  the  horizontal  joint  from  the  cen- 

troid  to  the  point  locating  the  neutral  axis. 
£  =  length  of  the  horizontal  joint. 
Cmax  =  maximum  compressive  stress  on  the  joint. 
Tmax=  maximum  tensile  stress  on  the  joint. 


APPENDIX  I  55 

Q  =  vertical   component   of   the   resultant  force   acting 

on  the  joint. 
A  =  area  of  the  joint. 
5  =  shear  at  any  point  y  in  the  joint. 
P  =  total  shear  on  the  joint. 
y  =  the  distance  from   the  centroid   to  any   point   on 

the  joint. 

With  regard  to  Eq.  (4)  it  may  be  stated  that  it  has 
not  heretofore  been  customary  to  consider  the  distribution 
of  shearing  stress  on  horizontal  joints.  But,  if  the  dis- 
tribution of  normal  stresses  may  be  assumed  to  be  repre- 
sented by  Eqs.  (i),  (2),  and  (3),  with  equal  validity  for 
the  usual  types  of  dam,  may  the  shear  at  any  point  be 
assumed  to  be  represented  by  Eq.  (4).  It  is  believed  by 
Mr.  Atcherley  that  these  equations  more  nearly  express  the 
conditions  of  equilibrium  in  a  dam  than  the  usual  ones  do, 
even  though  the  latter  tacitly  assume  the  first  three  by 
imposing  the  condition  of  the  middle  third,  and  use  a  fric- 
tion condition,  instead  of  one  for  shear  as  expressed  by 
Eq.  (4). 

In  reference  to  this  friction  .actor  there  may  be  some 
question  of  doubt,  since  M.  Levy*  prescribes  an  angle 
of  30°  for  masonry  on  masonry,  while  Rankine  gives  36°; 
on  the  other  hand,  examination  of  dams  actually  built 
frequently  shows  the  angle  to  lie  somewhere  between  the 
above  values. 

But  whatever  its  exact  value,  the  friction  condition 
leaves  some  doubt  as  to  the  actual  distribution  of  shear 

*  "  La  Statique  graphique."  IV®  Partie,  '  Ouvrages  en  Maf  onne- 
rie,"  page  92. 


56  HIGH    MASONRY    DAM    DESIGN 

over  a  horizontal  joint,  the  variation  of  which  must  be 
known,  in  order  to  determine  the  tensile  and  compressive 
stresses  on  the  vertical  sections  of  the  tail  (i.e.,  downstream 
portion)  of  the  dam.  In  consequence  of  this  the  parabolic 
law  as  expressed  by  Eq.  (4)  has  been  assumed  and  will 
later  be  shown  to  be  more  nearly  correct  than  any  other 
hypothesis. 

According  to  the  author  there  is  no  reason  whatever 
why  dams  should  be  tested  solely  by  taking  horizontal 
cross-sections,  and  asserting  that  the  line  of  resistance 
must  lie  in  the  middle  third,  while  the  stresses  across 
the  vertical  sections  of  the  tail  are  absolutely  neglected. 
If  the  former  condition  is  valid,  then  no  dam  ought  to 
be  passed  unless  it  can  be  shown  also  that  there  is  no  ten- 
sion of  any  serious  value  across  vertical  cross-sections  of 
the  tail,  parallel  to  the  length  of  the  structure.  It  is 
believed  that  a  great  number  of  dams  as  now  designed 
will  be  found  to  have  very  substantial  tension  in  these 
sections  and  this,  in  the  opinion  of  the  author,  is  a  source 
of  weakness  .in  dam  construction  which  has  not  been 
properly  considered  and  allowed  for. 

If  the  problem  is  to  be  solved  on  the  assumption  that 
a  dam  is  an  "  isotropic  and  homogeneous  "  structure,  the 
general  equations  for  the  stresses  can  be  determined  only 
by  the  following  considerations: 

(a)  The  normal  and  shearing  stresses  on  the  horizontal 
top    and   curved   flank,   i.e.,   downstream    face,   are    both 
zero. 

(b)  The   normal   stress    on   the   battered  front   or  up- 
stream face  is  equal  to  the  water  pressure,  and  the  shear  is 
zero,  and 


APPENDIX  I  57 

(c)  Either  the  stresses  or  the  shifts  must  be  supposed 
given  over  the  base. 

It  follows  at  once  from  this  that  Eqs.  (i),  (2),  and  (3) 
are  not  absolutely  true,  but  that  the  shear  is  fairly  closely 
represented  by  Eq.  (4). 

As  far  as  the  present  investigation  is  concerned,  however, 
the  enquiry  is  not  as  to  the  validity  of  the  usual  treatment; 
it  is  obviously  faulty.  But  it  is  the  purpose  to  try  to 
indicate  that,  supposing  it  to  be  correct,  its  present  partial 
application,  i.e.,  to  horizontal  joints  only,  involves  the 
serious,  and,  it  is  believed,  often  dangerous,  neglect  of 
large  tension  across  the  vertical  sections. 

To  justify  the  above  statement,  two  model  dams  of 
wood  were  employed  for  experimental  purposes,  the  cross- 
sections  being  identical,  and  agreeing  with  that  of  a  dam 
actually  constructed.  One  of  these  models  was  sub- 
divided into  horizontal  strata  to  study  the  effect  on  such 
planes,  and  the  other  into  vertical  longitudinal  strata, 
for  a  similar  purpose.  The  application  of  the  loading 
was  such  that  it  approximated  as  closely  as  possible  the 
conditions  obtaining  in  an  actual  dam.  The  general  con- 
clusions from  these  experiments  were  that: 

(a)  The  current  idea  that  the  critical  sections  of  a  dam 
are   the   horizontal   ones    is    entirely   erroneous.     A   dam 
collapses  first  by  the  tension  on  the  vertical  sections  of 
the  tail. 

(b)  The   shearing   of   the   vertical   sections   over   each 
other  follows  immediately  on  this  opening  up  by  tension. 

(c)  It  is   probable   that   the   shear  on   the  horizontal 
sections  is  also  a  far  more  important  matter  than  is  usually 
supposed. 


58  HIGH    MASONRY    DAM    DESIGN 

It  follows  consequently,  that  keeping  the  line  of  resist- 
ance within  the  middle  third  of  the  horizontal  sections 
is  by  no  means  the  hardest  part  of  dam  design.  It  would 
be  surprising  if,  with  all  the  labor  spent  on  this  point, 
the  bulk  of  existing  dam  constructions  are  not,  for  masonry, 
under  very  considerable  tension,  i.e.,  a  tension  across  the 
vertical  sections  'which  has  been  hitherto  disregarded. 

It  is  proposed  therefore  to  lay  it  down  as  a  rule  for  the 
construction  of  future  dams  that  the  stability  of  the  dam 
from  the  standpoint  of  the  vertical  sections  must  be  con- 
sidered in  the  first  place.  If  this  be  satisfactory,  it  is 
believed  that  the  horizontal  sections  will  be  found  to  be 
stable,  but  of  course  the  latter  must  be  independently 
investigated. 

The  above  conclusions  were  apparently  verified  by  a 
combined  analytical  and  graphical  treatment  in  which 
the  algebraical  analysis  will  here  be  considered  first. 

Denoting  the  total  vertical  force  acting  on  a  horizontal 
joint  by  Q0,  and  the  total  horizontal  force  acting  over  the 
same  by  P0,  under  the  assumption  that  the  reservoir  is 
full,  the  variation  of  the  normal  pressure  on  the  joint 
may  be  represented  by  the  straight  line  of  Eq.  (i). 

If  the  resultant  pressure  on  the  joint  be  assumed  to 
cut  it  at  the  extremity  of  the  middle  third,  then  according 

b2 

to  the  previous  notation,  d  will  have  a  value  of  J— ,  pro- 
vided 2b  is  the  length  of  the  joint.  This  indicates  that 
the  line  representing  the  variation  of  normal  pressure 
over  the  joint  intersects  it  at  the  upstream  edge,  and  any 
vertical  between  it  and  the  joint  itself  will  represent  the 
normal  pressure  at  that  point  where  the  vertical  is  erected. 


APPENDIX  I  59 

Denoting  this  by  y,  it  may  be  termed  "  the  vertical  height- 
giving  pressure,"  and  may  also  be  expressed  in  terms  of 
height  of  masonry,  if  the  factors  upon  which  it  depends 
are  expressed  in  cubic  feet  of  masonry. 

Again,  we  may  write  an  equation  of  the  downstream 
face,  with  respect  to  the  same  joint  so  long  as  that  face 
is  a  straight  line,  by  making  y'  =mx. 

Evidently  then  if  this  latter  line,  and  the  one  indicating 
the  variation  of  pressure  over  the  base,  be  referred  to 
the  same  origin,  the  tip  of  the  tail,  the  difference  in  areas 
included  between  each  and  the  base  will  represent  the 
total  upward  force,  in  cubic  feet  of  masonry,  acting  over 
any  assumed  portion,  "  x  "  of  the  joint,  measured  from 
the  tail. 

Representing  this  upward  force  by  FI  its  point  of 
application  may  be  easily  determined,  while  the  shear 
may  be  written  as  F2,  being  regulated  by  Eq.  (4). 

As  F\  and  F%  thus  give  all  the  external  forces,  con- 
sidering a  wedge-shaped  piece  of  dam  bounded  by  the 
downstream  face,  a  vertical  and  a  horizontal  plane,  the 
total  shear  on  the  vertical  plane  must  equal  F\  and  the 
total  thrust  F2,  since  these  internal  stresses  are  held  in 
equilibrium. by  external  forces.  Thus  F\  equals  the  total 
shear  on  the  vertical  section,  at  a  distance  x  from  the 
tip  of  the  tail,  while  F2  equals  the  total  horizontal  thrust 
over  the  same. 

If  y  be  expressed  in  terms  of  x,  and  locate  the  point  on 
the  successive  vertical  planes  through  which  the  resultant 
acts,  then  the  equation  will  represent  the  line  of  resistance 
on  these  vertical  planes.  It  is  found  to  be  an  hyperbola. 

Considering  the  stresses  on  the  vertical  sections,  it  is 


60  HIGH    MASONRY    DAM    DESIGN 

found:  First,  that  the  maximum  shear  may  be  properly 
represented  by  f  the  mean  value,  and  may  be  so  arranged 
as  to  be  expressed  in  terms  of  F\  and  mx.  Such  an  equa- 
tion, representing  a  straight  line,  immediately  shows  the 
necessity  of  thickening  the  tip  of  the  tail  which,  as  a 
matter  of  fact,  is  the  usual  procedure  in  actual  design. 
Second,  the  line  representing  the  maximum  tensile  stress 
may  be  shown  to  vary  as  a  parabola  whose  axis  is  vertical. 
When  the  downstream  face  ceases  to  be  linear,  it 
becomes  necessary  to  apply  a  graphical  solution  for  the 
determination  of  the  stresses.  This  it  is  unnecessary  to 
reproduce  here,  but  the  curves  may  be  said  to  indicate 
the  following  results: 

(1)  That  the  line  of  resistance  for  the  vertical  sections 
lies  outside  the  middle  third  for  rather  more   than  half 
the  vertical  sections.     In  other  words,  these  sections  are 
subjected  to  tension. 

(2)  That  the  tensile  stresses  in  the  tail  are,  for  masonry, 
very  serious,  c  mounting  to  nearly  10  tons  per  square  foot 
at  the  extreme  tip,  and  to  6  tons  per  square  foot  after  we 
have  passed  the  vertical  section,  where  the  strengthening 
of  the  tail  has  ceased. 

(3)  That  the  maximum  shearing  stresses  amount  to  6 
tons  per  square  foot  at  the  tip  of  the  tail  and  5  tons  per 
square  foot  after  we  have  passed  the  vertical  section,  where 
the  strengthening  of  the  tail  has  ceased.     No  undue  import- 
ance should  be  laid  on  the  actual  values  of  these  "  maxi- 
mum ' '  shears  on  the  vertical  sections  however,  as  they  are 
obtained  from  the  mean  shears  by  using  the  round  multi- 
plier   1.5.     This   round   number   is   assumed   because   the 
maximum  is  certainly  greater  than  the  mean  shear.     The 


APPENDIX  I  61 

actual  distribution  of  shear  on  the  vertical  sections  has 
not  been  discussed.  It  could,  of  course,  be  found  from 
that  on  the  horizontal  sections,  if  the  latter  were  really 
known  with  sufficient  accuracy,  by  the  equality  of  the 
shears  on  two  planes  at  right  angles.  It  is  sufficient  to 
show  that  the  mean  shears  on  the  vertical  sections  appear 
to  be  higher  than  those  on  the  horizontal  section,  and 
thus  indicate  that  the  parabolic  distribution  applied  to 
sections  some  way  above  the  base,  probably  under-esti- 
mates  the  maximum  shearing  in  the  dam. 

In  other  words:  Whether  the  test  is  made  by  the  line 
of  resistance  lying  outside  the  middle  third,  or  by  the  ex- 
istence of  serious  tensile  stresses,  or  by  the  magnitude  of 
the  mean  shearing  stresses,  the  vertical  sections  are  critical 
for  the  stability  in  a  far  higher  degree  than  the  horizontal 
sections. 

In  a  well-designed  dam,  all  the  conditions  for  stability 
of  the  horizontal  sections  may  have  been  satisfied,  yet 
if  the  very  same  conditions  be  applied  to  the  vertical 
sections  not  one  of  them  will  be  found  to  be  satisfied. 
It  seems  accordingly  very  unsatisfactory  that  the  current 
tests  for  stability  should,  if  they  are  legitimate,  be  applied 
to  the  horizontal  instead  of  to  the  far  more  critical  vertical 
sections.  In  the  case  of  the  latter  they  fail  completely; 
and  if  higher  tension  and  shear  are  to  be  allowed  in  the 
vertical  sections,  then  it  is  absurd  to  exclude  them  in  the 
case  of  the  horizontal  sections.  It  is  maintained  by  the 
author  that  the  current  treatment  of  dams  is  fallacious, 
for  it  screens  entirely  the  real  source  of  weakness,  namely, 
in  the  first  place  the  tension,  and  in  the  second  place  the 
substantial  shear,  in  the  vertical  sections,  and  this  at  dis- 


62  HIGH    MASONRY    DAM     DESIGN 

tances  from  the  tail  far  beyond  the  usual  tail-strengthen- 
ing range. 

Nor  do  these  theoretical  results  stand  unverified  by 
experiment;  they  are  absolutely  in  accord  with  the  ex- 
periments on  the  model  dams.  These  collapsed  precisely 
as  might  have  been  expected  from  the  above  investiga- 
tion, i.e.,  the  dam  with  vertical  sections  gave  long  before 
the  dam  with  horizontal  sections.  The  former  collapsed 
by  opening  up  of  the  joints  by  tension  towards  the  tail, 
followed  almost  immediately  by  a  shear  of  the  whole 
structure.  In  the  case  of  the  horizontally  stratified  dam, 
the  collapse,  which  occurred  much  later,  was  by  shear  of 
the  base,  followed  almost  simultaneously  by  a  shear  of 
one  or  more  of  the  horizontal  sections. 

The  question  then  arises  as  to  how  far  the  previously 
assumed  distribution  of  shear  affects  the  main  features 
of  the  results,  and  so  the  other  extreme  was  taken,  i.e., 
uniform  shear,  and  the  effect  determined. 

This  distribution  must  be  further  from  the  actual 
than  the  first  hypothesis,  yet  it  is  still  found: 

(1)  That  the  line  of  resistance  falls  well  outside  the 
middle  third  for  about  half  the  dam. 

(2)  That  there  exist  considerable  tensions,  3  to  4  tons 
per  square  foot,  in  the  masonry. 

(3)  That  the  average  shearing  stresses  on  the  vertical 
sections  are  greater  than  on  the  horizontal  sections.     As  a 
result  of  this  extreme  case,   it  is  believed  that  the  real 
distribution  of  shear  over  the  base,  whatever  it  may  be, 
must  lead  us  to  a  line  of  resistance  lying  well  outside  the 
middle   third,   and   to   tensions   amounting   to   something 
between  5  and  10  tons  per  square  foot. 


APPENDIX  I  63 

From    these    investigations    the    author   concludes    as 
follows : 

(1)  The  current  theory  of  the  stability  of  dams  is  both 
theoretically  and  experimentally  erroneous,  because : 

(a)  Theory  shows  that  the  vertical  and  not  the  hori- 
zontal sections  are  the  critical  sections. 

(b)  Experiment  shows  that  a  dam  first  gives  by  tension 
of  the  vertical  sections  near  the  tail. 

(2)  An  accepted  form  of  cross-section  is  shown  to  be 
stable  as  far  as  the  horizontal  sections  are  concerned,  but 
unstable    by    applying    the    same    conditions    of    stability 
to  the  vertical  sections. 

(3)  The  distribution  of  shear  over  the  base  must  be 
more  nearly  parabolic  than  uniform,   but  as  no  reversal 
of  the  statements  follows  in  passing  from  the  former  to 
the  latter  extreme  hypothesis,  it  is  not  unreasonable  to 
assume  the  former  distribution  will  describe  fairly  closely 
the  facts  until  we  have  greater  knowledge. 

(4)  In  future  it  is  held  that  in  the  first  place  masonry 
dams  must  be  investigated  for  the  stability  of  their  vertical 
sections.     If  this  be  done  it  is  believed  that  most  existing 
dams   will   be   found   to   fail,    if   the   criteria   of   stability 
usually  adopted  for  their  horizontal  sections  be  accepted. 
This  failure  can  be  met  in  two  ways : 

(a)  By  a  modification  of  the  customary  cross-section. 
It  is  probable  that  a  cross-section  like  that  of  the  Vyrnwy 
dam  would  give  better  results  than  more  usual  forms. 

(b)  By  a  frank  acceptance  that  masonry,  if  carefully 
built,  may  be  trusted  to  stand  a  definite  amount  of  tensile 
stress.     It  is  perfectly  idle  to  assert  that  it  is  absolutely 
necessary  that  the  line  of  resistance  shall  lie  in  the  middle 


64 


HIGH    MASONRY    DAM    DESIGN 


third  for  a  horizontal  treatment,  when  it  lies  well  out- 
side the  middle  third  for  at  least  half  the  dam  for  a 
vertical  treatment. 

Immediately  upon  the  publication  of  the  preceding 
results,  Sir  Benjamin  Baker  undertook  some  experiments 
of  a  like  nature.*  The  models  employed  by  him  were 


Radius  =Infinity 


__.*._ 


117.75 
Vyrnwy  Dam 

FIG.  i. 


made  of  ordinary  jelly  however,  and  included  not  only 
the  transverse  section  of  the  dam  itself,  but  the  rock  upon 
which  it  rested  as  well.  It  is  shown  in  the  figure. 

The  horizontal  and  vertical  lines  drawn  on  the  sides 
of  the  model  were  for  the  purpose  of  detecting  any  dis- 
tortion that  might  result  through  the  application  of 


*  Vol.  162,  page  120.     Minutes  of  Proceedings  of  the  Institution  of 
Civil  Engineers. 


APPENDIX  I 


65 


pressure.  These  pressures  were  applied  against  both 
the  upstream  face  and  the  floor  of  the  reservoir,  as  it 
was  believed  that,  while  according  to  the  theory  of  the 
middle  third  there  could  be  no  tension  in  the  heel,  never- 
theless for  the  case  of  reservoir  full,  fairly  severe  tension 
in  the  masonry  might  thus  be  caused. 

The    experiments    indicated    that    the    distribution    of 
shearing  stress  in  the  plane  of  the  base,  i.e.,  where  the 


FIG.  2. 


dam  met  the  rock,  was  more  nearly  uniform  than  para- 
bolic, and  that  the  strain  extended  into  the  rock  for  a 
distance  equal  to  about  half  the  height  of  the  dam  before 
it  became  undetectable.  To  solve  the  complete  problem, 
therefore,  it  would  be  necessary  to  consider  the  elasticity 
.of  the  rock  on  which  the  dam  rested.  Partially  as  a  result 
of  these  and  the  previous  experiments,  it  may  be  pointed 
out  in  passing,  the  proposed  increase  in  elevation  of  the 
Assouan  dam,  whereby  the  capacity  of  the  reservoir 


66  HIGH    MASONRY    DAM    DESIGN 

would  have  been  considerably  augmented,  was  indefinitely 
postponed. 

The  new  feature  in  Atcherley's  analysis  is  that,  even 
though  the  condition  of  "no  tension  in  a  horizontal 
joint  "  is  satisfied,  dangerous  tensions  may  be  shown  to 
exist  across  vertical  planes.*  In  connection  with  this 
consider,  for  example,  a  section  of  the  dam  ABC,  which 
is  triangular  in  profile,  and  construct  BEC  so  that  the 
ordinates  represent  the  variation  of  the  unit  normal 
stress  over  the  horizontal  joint  BC. 

Taking  a  vertical  section  IK  in  which  /  locates  the 
centroid,  the  forces  to  the  left  are  the  upward  pressure 
acting  over  BK,  tending  to  cause  rota- 
tion in  a  clock-wise  manner  and  thus 
produce  tension  at  K,  and  two  counter- 
acting forces  tending  to  neutralize  this 
pressure :  the  weight  of  the  portion  BKH 
and  the  horizontal  shearing  force  acting 
along  BK.  The  resultant  effect  of  all 
three  will  be  tension  at  K,  provided  the 
rotation  is  right-handed,  with  a  conse- 
quent splitting  along  the  vertical  plane  HK. 

In  view  of  the  fact  that  the  horizontal  shear  is  present 
as  a  factor,  it  is  necessary  to  determine  its  distribution, 
and  this  Prof.  W.  C.  Unwin  undertook  to  do.f  Instead 
however,  of  accepting  the  distribution  in  accordance  with 
Atcherley's  assumptions,  an  analysis  was  attempted  by 


*"  Engineering,"  Vol.  79,  page  414. 

f"  Engineering,"  Vol.  79,  page  513.     "Note  on  the  Theory  of  Un- 
symmetrical  Masonry  Dams,"  by  W.  C.  Unwin. 


UNIVERSITY 


APPENDIX   I 


67 


which  the  shear  might  be  actually  calculated,  and  in 
doing  so  attention  was  called  to  the  fact  that  the  accepted 
theory  of  dam  design  is  incomplete  in  just  that  feature, 
since  it  fails  to  consider  the  rate  of  change  in  the  hori- 
zontal shear. 

In  any  analysis  the  fundamental  assumption  must  be 
made  that  a  masonry  dam  is  a  homogeneous-elastic  solid, 
and,  while  it  is  not  absolutely  essential  that  no  tension 
exist  at  any  point  in  the  cross-section,  yet  it  seems  desir- 
able that  there  should  be  none  at  the  upstream  face  of 
horizontal  joints. 

It  may  be  said  therefore,  that  for  a  more  exact  analysis 
the  problem  resolves  itself  into  one  of  the  determination 
of  shear  on  horizontal  planes,  and  Prof.  Unwin  suggests 
as  follows,  a  method  of  procedure  by  which  this  may  be 
accomplished : 

If,  as  in  the  figure,  we  assume  a  dam  of  triangular 
section,  in  which  AB  is  some 
horizontal  joint,  other  than 
the  base,  and  C  its  centroid, 
then  Q  will  represent  the  water 
thrust,  P  the  weight  of  ma- 
sonry, and  R  their  resultant. 

In  agreement  with  the  or- 
dinary theory  we  may  write 
the  well-known  formula  for 
the  unit  normal  pressure  on  a 
horizontal  joint,  at  any  point  x,  measured  from  A,  as 
follows : 


FIG.  4. 


Pn=^r    1  + 


b2 


(5) 


68 


HIGH   MASONRY    DAM    DESIGN 


For  the  horizontal  shear  we  must  proceed  further. 
Consider  therefore,  the  forces  to  the  left  of  HK  in  Fig.  i 
we  have  (i)  the  vertical  pressure  qn  A K,  (2)  the  weight 
of  AHK,  and  (3)  the  shear  acting  along  AK.  It  is 
evident  that  the  difference  between  (i)  and  (2)  repre- 
sents the  total  vertical  shear  on  HK. 

If,  therefore,  the  figure  ALMB  represent,  in  masonry 
units,  the  distribution  of  normal  stress  on  AB,  as  given 
by  Eq.  (i),  then  ALTH  will,  in  like  manner,  represent 
the  above-mentioned  total  vertical  shear  on  HK. 


B' 


FIG.  5. 


JS      K 

FIG.  6. 


Consider  now  a  second  section  A'B1 ',  a  small  distance 
z  above  AB]  the  total  shear  on  HK'  may  then  be  found  as 
before.  Denoting  the  former  by  5,  and  the  latter  by 
5',  then  5  —  5'  equals  the  total  shear  on  KK' ,  which, 
when  divided  by  z,  will  give  the  intensity  of  vertical  shear 
at  K,  and  consequently  the  intensity  of  horizontal  shear 
at  the  same  point. 

Since  all  the  forces  to  the  left  of  HK  are  now  known, 
the  normal  stress  on  that  plane  may  be  found,  and  from 
it  we  may  readily  determine  whether  tension  or  com- 
pression exists  at  K. 

At  the  base  these  results  would  be  much  modified, 


APPENDIX   I  69 

because  of  the  discontinuity  of  form,  which,  in  the  opinion 
of  Prof.  Unwin,  places  the  exact  determination  of  the 
stresses  beyond  the  power  of  mathematics.  The  author 
believes  the  effect  of  the  rock  into  which  the  dam  is  built 
is  to  reduce  the  variation  of  stress  which  would  otherwise 
exist. 

In  a  subsequent  paper,*  giving  a  complete  demon- 
stration of  the  preceding  analysis  as  applied  to  a  masonry 
dam  of  triangular  cross-section,  it  is  found  that  the  dis- 
tribution of  shear  on  a  plane  horizontal  joint  may  be 
represented  by  a  right  triangle  whose  base  is  the  length  of 
the  joint  and  whose  vertex  is 
perpendicularly  below  the  down- 
stream edge.  The  figure  illus- 
trates the  variation  of  normal 
stress  and  shear  on  AB\  the 
lines  of  resistance  for  both 
vertical  and  horizontal  planes; 
and  the  centers  of  gravity  of 
the  sections  above  the  successive 
horizontal  joints. 

Consequently  the  total  nor- 
mal or  shearing  stress  on  any 
part  of  AB  is  equal  to  the  area 
between  that  part  and  the  line 

of  normal  stress  or  the  line  of 

FIG.  7. 

shearing  stress. 

If  the  upward  reactions  and  the  weights  of  the  dam  to 


*"  Engineering,"  Vol.  79,  page  593.     "Further  Note  on  the  Theory 
of  Unsymmetrical   Masonry  Dams."     W.  C.  Unwin. 


70  HIGH    MASONRY    DAM   DESIGN 

the  left  of  each  vertical  section  be  combined  with  the 
shears  T,  acting  along  AB,  the  resultants  will  cut  the 
vertical  sections  at  points  shown  on  the  line  of  resist- 
ance for  these  vertical  sections.  As  this  line  lies  wholly 
within  the  middle  third,  there  can  be  no  tension  on  any 
vertical  section. 

The  total  compressive  stress   on  any  vertical  section 
at  its  lower  edge  will  therefore  be: 


(6) 


where  T  is  the  shear  on  the  horizontal  plane  from  the  toe 
to  the  vertical  section  taken,  y  the  height  of  the  vertical 
section,  and  z  the  distance  from  the  center  of  the  vertical 
section  to  the  point  of  application  of  the  resultant  forces 
on  that  section. 

Near  the  upstream  toe  the  plane  on  which  the  greater 
principal  stress  acts  is  found  to  be  vertical  while  near 
the  downstream  toe  it  approaches  the  horizontal.  The 
stresses  are  all  compressive  and  on  the  water  face  the 
compressive  stress  it  at  all  points  equal  to  the  water 
pressure  at  that  point. 

The  above  analysis  is  simply  an  application  to  vertical 
sections  of  the  method  now  accepted  as  applicable  to  the 
horizontal  planes  and  is  a  possible  solution,  since  the 
distribution  of  shear  is  known.  It  differs  from  Atcherly's 
method  in  the  fact  that  the  latter  assumes  the  usual 
distribution  of  normal  stress,  together  with  a  parabolic 
variation  for  the  horizontal  shear.  This  latter  hypothesis 
the  author  thinks  inconsistent  with  the  previous  one. 


APPENDIX  I  71 

Further  investigations  by  Prof.  Unwin*  on  dams  of 
various  sections  lead  to  the  following  conclusions: 

(1)  For  a  rectangular  dam  the  distribution  of  shearing 
stress   on   horizontal   planes   may   be   represented   by   the 
ordinates  of  a  parabola. 

(2)  For    a    triangular    dam,    the    distribution   may    be 
represented  by  the  ordinates  of  a  triangle  with  the  apex 
below  the  downstream  toe. 

(3)  For  a  dam  with  vertical  upstream  face  and  curved 
downstream    face    the    distribution    may    be    represented 
by   a    figure    consisting   of    a    parabola    superposed    on    a 
triangle. 

(4)  For  a  dam  with  rectangular  base  the  distribution 
is  represented  by  a  parabola. 

Following  the  results  of  the  experimental  investigations 
of  Atcherley  and  Baker,  several  other  papers  of  a  like 
nature  appeared  in  the  Minutes  of  Proceedings  of  the 
Institute  of  Civil  Engineers,  Vol.  162.  The  first  of 
these  to  be  considered  here  is  that  by  Sir  John  Walter 
Ottley  and  Arthur  William  Brightmore,  entitled,  "  Ex- 
perimental Investigations  of  the  Stresses  in  Masonry  Dams 
subjected  to  Water- Pressure." 

In  presenting  this  paper,  the  authors  drew  attention 
to  the  fact  that  until  the  publication  of  Mr.  Atcherley's 
results,  the  question  of  dam  design  had  been  accepted 
as  settled,  and  that  his  memoir  had  had  the  effect  of 
reopening  the  entire  subject  of  the  distribution  of  stress 
in  structures  of  this  class. 

*l< Engineering,"  Vol.  79,  page  825.  "On  the  Distribution  of 
Shearing  Stress  in  Masonry  Dams."  Prof.  W.  C.  Unwin. 


72  HIGH  MASONRY   DAN    DESIGN 

It  was  also  pointed  out  that  tension  was  found  by  him 
to  exist  on  vertical  planes  near  the  outer  toe,  whether 
the  distribution  of  shearing  stress  over  the  base  was 
assumed  to  be  uniform  or  to  vary  according  to  the  para- 
bolic law. 

Considering  a  transverse  section  of  a  dam,  the  authors 
argued  that,  whatever  the  distribution  of  shear  over  the 
base  might  be,  it  must  follow  some  other  law  near  the  top, 
since  the  conditions  in  these  higher  levels  are  radically 
different  from  those  existing  in  the  lower,  where  the  dam 
is  fixed  to  the  foundation,  and  where  the  water  pressure 
ceases  abruptly. 

The  investigation  was  therefore  undertaken,  at  least 
in  part,  to  determine  the  distribution  of  shear  on  horizontal 
planes  in  the  higher  levels  of  the  dam  and  to  see  how  it 
varied  from  that  at  the  base;  and  it  might  be  stated  here 
that  it  was  found  to  be  uniform  in  the  latter  plane  but  to 
vary  uniformly  from  zero  at  the  heel  to  a  maximum  at 
the  toe  in  the  higher  levels,  the  change  from  the  one 
condition  to  the  other  being  gradual.  It  will  be  shown 
that  it  is  near  the  inner  toe  rather  than  near  the  outer 
toe  that  tension  may  be  anticipated. 

The  model  dams  were  triangular  in  section,  made  from 
a  kind  of  modeling  clay  called  "  plasticine,"  and  so  pro- 
portioned that  the  resultant  pressure  on  the  base  cut 
that  plane  at  the  downstream  extremity  of  the  middle 
third. 

For  purposes  of  observation  the  sections  were  placed 
between  vertical  sides  of  plate  glass,  upon  which  vertical 
and  horizontal  lines  had  been  etched,  corresponding  to 
similar  lines  on  the  model,  so  that  any  displacement  in 


APPENDIX  I  73 

the  latter  might  be  noted  by  comparison  with  the  former. 
Pressure  was  applied,  by  means  of  a  thin  rubber  bag  con- 
taining water  which  was  made  to  fit  the  frame.  Though 
the  water  was  allowed  to  act  over  a  period  of  33  days, 
after  the  elapse  of  one  week  a  crack  was  noticed  at  the 
upstream  toe,  running  downward  and  at  an  angle  of 
about  45°.  At  the  end  of  the  longer  period  an  examination 
showed  that  in  the  neighborhood  of  the  base  the  dis- 
placement of  the  vertical  lines  was  such  as  to  make  them 
all  about  equally  inclined,  thus  indicating  a  uniform 
intensity  of  shear  on  that  section,  while  in  the  higher 
levels  and  near  the  outer  portion  of  the  dam  the  lines 
became  more  inclined  as  the  elevation  increased,  indicating 
that  the  intensity  of  shear  increased  also  as  the  top  was 
approached. 

Turning  to  the  horizontal  lines  in  the  model  for  the 
purpose  of  discovering  the  method  of  distribution  of 
normal  stress,  it  was  found  that  they  were  curves  at  the 
base,  sloping  downward  from  the  inner  toe  to  a  point 
about  two-thirds  the  distance  to  the  outer  toe,  then  re- 
maining fairly  level  until  almost  reaching  the  down- 
stream face,  when  they  finally  bent  up  slightly.  In  the 
higher  levels,  however,  these  lines  gradually  developed  a 
uniform  slope  running  from  the  inner  to  the  outer  toe. 

An  investigation  of  the  shearing  stresses  on  vertical 
planes  requires  that,  to  draw  the  line  representing  the 
intensity  of  normal  reaction  at  the  base  the  following 
facts  must  be  considered: 

(1)  The  total  normal  reaction  equals  the  weight  of  the 
dam. 

(2)  Since  the  resultant  pressure  on  the  base  acts  at 


74 


HIGH   MASONRY   DAM   DESIGN 


one-third  the  width  from  the  outer  toe,  the  moment  of  the 
reaction  stresses  about  this  point  must  be  zero. 

(3)  The  intensity  of  the  reaction  at  the  outer  toe 
must  equal  the  intensity  of  the  shearing  stress  in  the  vertical 
plane  multiplied  by  the  ratio  of  the  height  to  the  base  of 
the  dam. 




Averagelint^nsity  of  \Shearing^^    Stress  on 


FIG.  8. 


Referring  to  the  figure:  AB  represents  the  base  of  the 
dam,  and  EC  twice  the  average  intensity  of  normal  stress 
on  AB.  AC  is  then  drawn;  consequently  ABC  represents 
the  total  normal  stress  on  A  B,  or  the  weight  of  the  structure. 


APPENDIX  I  75 

If  AE,  on  the  other  hand,  represents  the  actual  intensity 
of  normal  reaction  over  AB,  then  for  (i)  to  hold  true  the 
area  Y  must  equal  the  areas  (x  +  z)  and  if  (2)  is  to  hold, 
the  moments  of  x,  y,  and  z,  about  D  (equal  to  %AB  from 
B),  must  be  zero;  also  for  (3)  to  be  satisfied,  BE  must 
equal  the  limiting  value  of  shearing  stress  in  a  vertical 
plane  near  the  toe,  multiplied  by  the  height  and  divided 
by  the  base  of  the  dam. 

From  these  considerations  AE  may  be  fitted  in  by 
trial  till  it  is  found  to  satisfy  all  of  the  above  conditions. 

Dividing  the  cross-sections  into  strips  i  inch  wide 
we  may  properly  consider  the  equilibrium  of  each  such 
strip.  Evidently  the  difference  between  the  weight  of 
each  strip  and  the  normal  reaction  on  the  base  is  equal 
to  the  difference  in  shear  on  the  two  adjacent  vertical 
planes,  and  if  in  the  figure  these  weights  be  plotted  upward 
from  AE,  the  curve  FE  will  result.  Furthermore,  both 
the  curves  for  "  total  shear  on  vertical  planes  "  and 
"  average  intensity  of  shear  on  vertical  planes  "  may  now 
be  drawn,  whereupon  it  is  evident  to  what  extent  the 
average  intensity  of  shear  on  vertical  planes  varies,  and 
how  it  compares  with  the  average  intensity  on  the 
base. 

Since  the  shear  on  horizontal  and  vertical  planes  at 
any  one  point  is  equal,  and  the  shear  on  the  base  is  practi- 
cally constant,  it  follows  that  above  the  base  the  shear  on 
horizontal  or  vertical  planes  is  small  near  the  heel  while 
in  the  outer  half  above  the  base  it  increases  as  the  outer 
edge  is  approached;  in  fact  it  increases  from  zero  at  the 
heel  to  a  maximum  at  the  toe.  These  facts  show  that  the 
shearing  stresses  to  be  provided  for  are  those  existing 


76  HIGH    MASONRY   DAM   DESIGN 

in  the  higher  levels  and  near  the  toe,  and  not  those  at  the 
base. 

In  considering  the  effect  of  shear  on  the  base,  neglecting 
the  "  fixing  "  at  that  level,  we  may  assume  that  the  re- 
action stress  and  that  due  to  the  weight  of  a  strip,  is  constant 
over  each  inch.  They  then  act  at  the  middle  of  each 
strip;  and,  taking  these  points  successively  as  centers, 
the  difference  of  the  moments  of  the  horizontal  pressures 
on  the  vertical  sides  of  the  strip,  it  is  evident,  will  equal 
the  sum  of  the  shearing  stresses  on  the  same  vertical  sides 
multiplied  by  J  inch. 

This  makes  possible  the  determination  of  the  moment 
of  the  horizontal  pressures  on  each  vertical  strip. 

The  horizontal  shear  on  each  inch  of  base  being  the 
difference  between  the  horizontal  pressures  acting  on  the 
two  vertical  sides,  the  latter  may  be  determined  as  soon 
as  their  points  of  application  are  given.  As  these  points 
are  known  for  the  innermost  and  outermost  strip,  an  easy 
curve  may  be  drawn  which  will  approximately  locate  the 
other  points  and  thus  give  the  desired  heights.  From 
these  results  it  may  be  shown  that  the  shearing  stress  on 
the  base  increases  from  practically  zero  at  the  inner  toe 
to  a  point  near  the  center  of  the  base  and  then  remains 
fairly  constant. 

The  modification  of  this  distribution,  due  to  the  fixing 
of  the  dam  to  its  base,  must,  on  the  other  hand,  be  con- 
sidered. The  water  tends  to  cause  a  maximum  pressure 
and  displacement  at  the  inner  face,  which  diminishes  to 
zero  at  the  outer.  As  the  dam  is  fixed,  this  displacement 
is  prevented,  thus  inducing  corresponding  shears,  and  the 
effect  of  this  conflicting  condition,  with  that  previously 


APPENDIX  I  77 

shown  to  exist,  causes  a  nearly  uniform  shear  over  the 
base. 

Further  evidence  of  uniform  shear  on  the  base  was 
obtained  as  follows:  The  models,  after  being  subjected 
to  water  pressure,  showed  cracks  which  appeared  at  the 
inner  toe,  the  angles  which  these  made  with  the  horizontal 
steadily  diminishing  as  the  base  was  decreased  in  width 
from  a  maximum  of  45°  for  the  widest  base  used  to  25° 
for  the  narrowest. 

The  variation  of  these  inclinations  corresponded  closely 
with  the  computed  directions,  on  the  assumption  that  the 
shear  was  uniform  over  the  base  and  the  experiments 
therefore  strongly  support  the  inference  that  shear  over 
the  base  is  uniformly  distributed. 

It  was  shown  by  means  of  the  models  that  there  are 
tensile  stresses  on  other  than  horizontal  planes  passing 
through  the  inner  toe.  The  models  indicated  this  by 
cracking,  even  when  the  back  was  sloped  away  from  the 
vertical  so  as  to  cause  vertical  pressure  and  hence  com- 
pression on  the  upstream  face. 

The  impossibility  of  tension  on  vertical  planes  near 
the  outer  toe  may  be  shown  by  means  of  the  following 
equation  for  principal  stress: 


where  compressions  are  plus  and  tensions  are  minus. 
When  pp'>(f  at  any  point,  there  can  be  no  tension  at 
that  point,  since  under  the  above  conditions  both  principal 
stresses  will  be  compression  and  hence  stresses  on  all  other 
planes  passing  through  that  point  will  be  compression 


78  HIGH   MASONRY    DAM    DESIGN 

also.  This  condition  may  be  shown  to  exist  near  the  outer 
toe,  and  hence  no  tension  can  act  across  any  vertical 
plane  in  that  position. 

For  example  consider  the  equilibrium  of  a  wedge  of 
unit  length  cut  off  by  a  vertical  plane  near  the  toe. 

p'  =  intensity  of  pressure  normal  to  a  vertical  plane 

at  base. 

p  =  intensity  of  reaction  normal  to  the  base. 
q  =  intensity  of  shearing  stress. 


Then  p'h  =  qb  or, 


(8) 


The  weight  of  the  particle  is  negligible  because  it  varies 
with  h2. 

Since  the  resultant  stress  must  be  parallel  to  the  outer 
face  it  follows  that, 


(9) 


Multiplying  (8)  by  (9)  there  results 

ppf  =  cf-  at  outer  toe.  But  p  has  been  shown  to  in- 
crease for  some  distance  from  outer  toe  and  the  point  of 
application  of  p'  becomes  relatively  lower  as  the  inner 
toe  is  approached-  and  since  the  average  pressure  is  con- 
stant it  follows  that  p  increases  as  the  distance  from  the 
outer  toe  increases  and  hence  in  the  vicinity  of  the  outer 
toe  pp'  is  greater  than  q2  and  consequently  there  can  be 
no  tension  in  that  neighborhood.  This  was  checked  by 
the  behavior  of  the  models. 


APPENDIX  I  79 

It  is  a  fact  that  in  dam  work  the  normal  stress  is  the 
only  one  specified,  whereas  the  absolute  maximum  is  about 
50  per  cent  greater. 

The  conclusions  reached  from  this  set  of  experiments 
follow : 

(1)  If  a  masonry  dam  be  designed  on  the  assumption 
that  the  stresses  on  the  base  are  uniformly  varying  and 
that  the  stresses  are  parallel  to  the  resultant  force  acting 
on  the  base,  the  actual  normal  and  shearing  stresses  on 
both  horizontal  and   vertical  planes  would  be  less   than 
those  provided  for. 

(2)  There  can  be  no  tension  on  any  planes  near  the 
outer  toe. 

(3)  There  will  be  tension  on  certain  planes  other  than 
the    horizontal    near    the    inner    toe,    and    the    maximum 
intensity  of  such  tension  in  the  foundation  being  generally 
equal  to  the  average  intensity  of  shearing  stress  on  the 
base,  and  the  inclination  of  its  plane  of  action  being  about 
45°;  and  its  maximum  intensity  in  the  dam  above  the  base 
about  J  the  above  amount  and  acting  on  a  plane  less  in- 
clined to  the  horizontal. 

The  investigation  undertaken  by  Mr.  Hill  *  for  "  The 
Determination  of  the  Stresses  on  any  Small  Element  of 
Mass  in  a  Masonry  Dam,"  are  on  the  other  hand  purely 
analytical  in  character,  being  directed  toward  a  solution 
of  (i)  the  vertical,  (2)  horizontal,  and  (3)  tangential  shearing 
forces  acting  on  the  faces  and  along  the  edges  of  such  an 
element. 

*  Minutes  of  Proceedings  of  the  Inst.  of  C.  E.,  72. 


80  HIGH    MASONRY   DAM   DESIGN 

In  this  analysis,  there  is  first  expressed  a  perfectly 
general  formula  for  C  (the  distance  of  the  load  point 
from  the  center  of  the  joint),  and  two  other  general  formulae 
for  the  pressures  pi  and  £2  in  terms  of  the  total  load  and 
C  from  its  above  value,  where  pi  is  the  minimum  and 
p2  the  maximum  pressure.  For  the  pressure  p  at  any 
point  oc  on  the  joint  of  length  b  the  following  equation 
is  used: 


(10) 


Up  to  this  point  the  analysis  is  identical  with  the  general 
procedure  of  investigation,  which  assumes  that  the  hori- 
zontal pressures  are  proportional  to  the  vertical,  and 
does  not  analyze  the  shear. 

Citing  Prof.  Unwin,  the  author  states  tnat  the  former 
"  suggested  that  the  shearing  stress  at  any  point  might 
be  found  by  considering  the  difference  between  the  total 
net  vertical  reactions  (between  that  point  and  either  face) 
along  two  horizontal  planes  a  unit's  distance  apart,  and 
has  applied  the  principle  by  the  use  of  algebraical  methods." 
Mr.  Hill,  on  the  contrary,  employs  the  calculus  to  obtain 
more  rigorous  results. 

The  procedure  follows:  Consider  any  point  distant  oc 
from  the  inner  toe  and  on  the  lower  of  two  horizontal 
planes,  a  unit's  distance  apart.  The  total  vertical  reaction 


is  then    I    pdoc.     Subtracting  the  weight  of  masonry  resting 
JQ 

on  this  portion  of  the  horizontal  joint,  and  denoting  the 
difference  by  r  we  have  an  expression  for  the  "  net 
vertical  reaction."  If  this  value  of  r  be  differentiated 


APPENDIX  I  81 

with  respect  to  h,  the  distance  between  the  two  hori- 
zontal planes,  the  change  in  the  reaction  will  be  obtained, 
and  this  change  or  difference  is  the  vertical  shearing 
stress  at  the  point  located  by  x.  It  is  also,  therefore,  the 
horizontal  shear  at  the  same  point,  which  we  may  denote 
byq. 

If  q  be  integrated  with  respect  to  x,  between  the  limits 
of  x  and  b,  the  resulting  expression  will  give  the  entire 
horizontal  shear  between  such  limits  on  the  joints  in 
question.  Represent  this  by  Qoc. 

To  find  the  horizontal  pressure  intensity,  we  have 
but  to  consider  the  above  integration.  This  shear  must 
be  resisted  by  the  material  along  the  vertical  section  at  x. 
Similarly  the  total  shear  on  a  plane  a  differential  distance 
below  the  last  must  be  resisted  by  the  vertical  section 
at  x,  differing  in  height  from  the  former  by  dh.  Conse- 
quently the  differential  of  Qoc  with  respect  to  h=p'  will 
represent  the  horizontal  pressure  intensity  at  point  x. 
These  expressions  for  pt  pf  and  q  therefore  give  respectively 
the  values  of  the  vertical  pressure  intensity,  horizontal 
pressure  intensity,  and  shearing  force  acting  on  a  unit 
element  of  mass. 

Cain  *  presents  a  treatment  of  this  matter,  which, 
while  presenting  no  new  features,  is  strictly  arithmetical 
in  character,  and  in  that  respect  at  least  differs  from  the 
preceding.  Its  purpose,  as  Hill's,  is  to  determine  -the 
amount  and  distribution  of  stress  at  any  point  in  a  masonry 


*  Wm.  Cain,  M.  Am.  Soc.  C.  E.,   Trans.   Am.   Soc.  C.   E.,   Vol.  64, 
page  208. 


82  HIGH   MASONRY    DAM    DESIGN 

dam,  on  the  assumption  that  the  law  of  the  trapezoid 
represents  the  variation  of  pressure  on  horizontal  joints. 

The  analysis  finally  establishes  formulae  for  (i)  the 
normal  unit  stress  at  any  point  in  a  horizontal  joint,  (2) 
the  normal  unit  stress  on  a  vertical  plane  at  any  point 
of  a  horizontal  joint,  (3)  the  unit  shear  on  either  horizontal 
or  vertical  planes  at  any  point  of  a  horizontal  joint,  and 
at  the  same  time  indicate  the  method  of  determining  the 
maximum  and  minimum  normal  stresses  and  the  planes 
on  which  they  act. 

The  solutions  are  only  approximate,  but  the  results  are 
found  to  be  close  enough  for  the  purpose. 

Before  proceeding  it  may  be  advisable  to  review  certain 
features  involved  in  a  consideration  of  the  stresses  in  a 
masonry  dam  which  Prof.  Cain  presents  in  a  very  satis- 
factory manner. 

i.  It  will  be  evident  from  an  examination  of  the  figure 
that  the  intensities  of  shear  on  two  planes  at  right  angles 
to  each  other  are  equal.       For,   in 
the  elementary  cube  under  consider- 
ation,  the  weight  may  be  neglected, 
—      since  it  is  an  infinitesimal  of  the  third 
order,    while    the    opposing    normal 
\,       I  forces  balance  as  the  cube  is  reduced 

in  size. 

For     equilibrium     then,    q-a-a  = 

q'-a-a,  or  q  =  qf  and,  because  each  side  is  a  differential 
quantity,  it  may  be  assumed  that  the  values  q  and  q' 
represent  the  average  unit  shear  on  the  respective  faces. 
As  a  consequence  they  are  equal  to  the  shear  at  any 
point,  for  example  A,  of  the  particle. 


APPENDIX  I 


83 


2.  In  a  triangular  element  of  the  dam,  at  the  down- 
stream edge,  and  of  unit's  length,  the  force  sacting'are 
those  shown.  Because  it  is  an  element  we  may  neglect 
the  weight,  and  therefore,  if  p'  is  the  normal  intensity  of 
stress  on  a  vertical  plane,  p  the  normal  intensity  of  stress 
on  a  horizontal  plane,  and  q  the  shear  intensity,  for 


for 


then 


or, 


or     p  =  q-r     and     g  = 


2H  =  o,  p'a  =  qb     or     p'  =  q  -. 


pf  =p  tan2  0, 


.        (12) 


p,  p  =  (f. 


(13) 


FIG.  10. 


FIG.  n. 


3.  The  same  analysis  may  be  applied  to  an  element 
at  the  inner  face,  where  $  is  the  inclination  to  the  vertical ; 
but,  for  the  reservoir  full,  the  intensity  of  water  pressure, 
horizontal  or  vertical,  at  c,  and  in  this  case  represented 
by  w,  must  be  taken  into  account. 


84  HIGH   MASONRY    DAM    DESIGN 

Under  these  circumstances, 

pb=qa  +  wb,     .     .     .     .     .     .     (14) 

and 

p'a=qb  +  wa,     .     .     .     ...     (15) 


.'.     p=qcot  <j>r  +w,  .....     (16) 
and 

(17) 


When,  as  is  usually  the  case,  the  vertical  component 
of  water  pressure  acting  along  the  back  is  neglected,  the 
above  equations  become, 

p  =  qcot  <£',     .     .     .    -.     .     .     (18) 
£'=<?tan  <£'  +  w,  .     *     .     .     .     (19)  . 

•'•     <?  =  £  tan  <£',    ......     (20) 

and 

(21) 


4.  If  an  element  at  the  down 
stream  face  be  again  considered,  since 
the  shear  on  the  outer  face  DC  is 
zero,  that  on  a  plane  AD  perpen- 
dicular  to  DC,  must  be  zero  also,  and 
hence  the  stress  d  on  AD  is  wholly 
FIG'  I2'  normal. 

The  total  pressure  on  AD  is  therefore, 

f'AD=f-bcos</>  .....     (22) 


APPENDIX  I  85 

The    vertical    component    of    this    is   /-6cos2<£,    because 


pb=f-bcos2  $, (23) 


'     '     '     (24) 


or, 


which  is  the  maximum  intensity  of  normal  stress  at  the 
outer  face. 


FIG.  13. 

5.  To  determine  the  planes  of  principal  stress,  i.e., 
planes  upon  which  the  stress  is  wholly  normal,  and  also 
the  intensity  of  that  stress,  we  may  assume  the  conditions 
indicated  in  the  figure. 

The  total  shear  on  c  then  is  fc ;  its  vertical  component 
fc.  cos  =/&,  and  its  horizontal  component  fc  sin  =fa. 

When  EV  =  o  and 


,    .     .     (25) 


f-p=qcot  0,         .     (26) 


86  HIGH   MASONRY    DAM    DESIGN 

The  difference  of  these  two  equations  gives, 

j  _  tan2  6 
p-p'=q(cotd-tan6)=q  .;     (27) 


2  tan  6          20 

-        '    '    (28) 


This  equation  gives  a  plane  upon  which  there  is  none 
but  normal  stress. 

To  determine/,  multiply  equation  (25)  by  (26). 

(/-/>)  (f-p')=q2>       ..  -   •  V   (29) 

whence, 

02-4(^'-<?2).     -     .     (30) 


This  will  give  two  values  of  /,  which  correspond  to  the 
two  principal  planes  of  stress,  the  stress  being  compressive 
when  /  is  positive,  and  tensile  when  /  is  negative.  There 
can  be  no  tension  when  p,  p'  >q2. 

Determination  of  the  vertical  unit  stress  at  any  paint 
of  a  horizontal  plane  joint:  From  the  law  of  the  trapezoid, 
we  have  the  pressures  at  the  upstream  and  downstream 
toes  represented  respectively  as  follows: 


T,. 

W,       .....     (31) 

w  ......     (32) 


The  resultant  is  supposed  to  act  within  the  middle 
third.  If  xr  represent  any  point  along  EB,  measured 
from  E,  then  p,  the  pressure  at  x,  is  given  by, 

p  =  p2  +  Plx,,         .....     (33) 


APPENDIX  I  87 

while  the  total  normal  stress  from  E  to  xf  is,  by  integration, 

^~TT x'2 (34) 


To  find  the  unit  shear  on  vertical  or  horizontal  planes, 
we  have  but  to  consider  a  slice  of  dam  between  two  hori- 
zontal joints  one  foot  apart,  extending  from  the  inner  to 
the  outer  face,  a  distance  x  along  the  lower  joint.  (The 
back  is  supposed  to  slope  .02  feet  for  each  foot  in  height). 


FIG.  14. 


The  vertical  forces  acting  are: 

(1)  A    uniformly    varying    stress    on    the    upper   joint 
acting  downward. 

(2)  The  same  on  the  lower  joint  acting  upward. 

(3)  The  weight  of  the  strip. 

(4)  The  shear  on  the  vertical  face  at  x. 
For  equilibrium, 

j.  -*•  '    \  /  \o  0  f 

Pf  and  P  may  be  obtained  as  indicated  in  the  previous 
demonstration. 


88  HIGH  MASONRY   DAM   DESIGN 

The  above  value  of  qi  is  the  average  unit  shear  at  the 
depth  taken,  but  a  similar  value  q2  may  be  determined 
at  a  depth  one  foot  below.  Under  these  circumstances 

— —  is  the  average  of  the  two,  and  may  be  said  to  be 
approximately  equal  to  the  shear  at  the  depth  of  the  joint 
between  the  two  slices. 


E 

a/—  o.oi 
a; 

H 

M 

~ 

—a 

/ 

a?  +0.01 

N 

FIG.  16. 

To  find  the  normal  unit  stress  on  a  vertical  plane,  a 
similar  section  to  that  just  used  may  be  employed;  but  the 
horizontal  components  are  now  to  be  equated  for  equilib- 
rium. 

Let  h  =  the  horizontal  water  pressure  at  the  assumed 

depth. 

Q'  =  total  shear  on  upper  face. 
Q  =  total  shear  on  lower  face. 
pf  =  average  normal  stress. 

qi  and  q2  =  the  intensities  of  horizontal  shear  at  the  points 
indicated. 

Q'  and  Q  may  be  found  by  integrating  qi  and  q2  be- 
tween the  proper  limits. 


-Q (36) 

This  value  of  pf  is  assumed  as  the  average  intensity 
on  the  vertical  plane  and  as  the  unit  intensity  on  the 


APPENDIX  I  89 

vertical  plane  at  a  point  midway  between  the  two  hori- 
zontal planes. 

Three  general  formulae  may  be  written  for  p3  q,  and  pf 
which,  it  has  been  suggested,  be  put  in  the  following  form: 

(37) 
(38) 
.     .     .     .     .     (39) 


APPENDIX  II 

THE   DESIGN  OF  A  HIGH  MASONRY  DAM 

THE  following  calculations  illustrate  the  method  em- 
ployed in  the  determination  of  the  theoretical  cross- 
section  of  a  high  masonry  dam. 

These  conditions  were  assumed: 

Height    of    free    water    surface    above    foundation, 

25o/-o//. 

Width  of  top,  23 '-o". 

Working  pressures:  p  =  i4  tons  per  square  foot. 
q  =  iS  tons  per  square  foot. 
Weight  of  masonry  146  pounds  per  cubic  foot.. 
Weight  of  water  62.5  pounds  per  cubic  foot. 
Coefficient  of  friction /  =  0.7. 

Joint  JL 

Generally  speaking  it  may  be  assumed  that  the  top 
of  the  dam  is  about  one-tenth  of  the  height  above  the 
water,  but  in  the  case  under  consideration,  a  supereleva- 
tion of  only  20  feet  will  be  employed.  While  the  choice 
in  this  respect  is  purely  arbitrary,  the  above  ratio  is  the 
one  usually  prescribed  if  there  are  no  other  governing 

conditions. 

90 


APPENDIX   II  91 

Since  the  length  of  a  joint  depends  'upon  its  depth 
below  the  water,  it  is  evident  that  at  the  surface  this 
dimension  should  be  zero.  For  various  reasons  however, 
such  as  the  desirability  of  a  footwalk  or  a  driveway  on 
the  crest,  a  top  width  is  chosen  which  will  satisfy  these 
demands. 

As  23  feet  has  been  decided  upon  in  this  problem, 
for  a  considerable  distance  below  the  water  level  the 
rectangular  section  will  more  than  satisfy  the  only  condition 
for  stability  that  applies  in  this  portion  and  which  re- 
quires that  the  resultant  of  all  the  external  forces  lie 
within  the  middle  third  of  the  cross-section.  It  becomes 
necessary  therefore,  to  determine  the  depth  at  which  a 
modification  of  this  dimension  should  take  place,  and 
for  this  purpose  Eq.  (34)  is  employed. 

(Series  A,  Stage  I.) 


,4  =  2.333,  L  =  23'.o,  a  =  2o'.o. 

This  equation  may  be  solved  by  successive  substitu- 
tions until  such  a  value  of  H  is  found  that  equality  results. 
In  the  present  instance  it  is  found  that  #  =  42.6  feet 
satisfies  the  equation,  and  hence  it  is  necessary  to  carry 
the  rectangular  cross-section  of  the  dam  down  to  a  depth 
of  62.6  feet  below  the  top. 

The  solution  may  be  expedited  by  the  use  of  the  graphic 
method.  Thus,  assume  at  least  three  values  for  H,  say 
30,  40,  and  50  feet  in  the  present  case,  substitute  suc- 
cessively in  the  right-hand  member  of  the  above  equation 
and  solve.  Plot  these  resulting  values  as  abscissae  and 


92  HIGH    MASONRY    DAM    DESIGN 

the  assumed  Values  of  H  corresponding  as  ordinates. 
A  smooth  curve  drawn  through  the  points  thus  obtained 
will  give  a  point  on  the  line  where  ordinate  and  abscissa 
are  equal,  and  this  will  be  the  desired  value. 

At  no  point  in  this  portion  of  the  dam  does  the  length 
of  a  horizontal  joint  change;  but  below  this  elevation 
the  dimension  will  have  to  be  increased  in  order  to  comply 
with  the  requirement  that  the  resultant  shall  not  pass 
outside  the  middle  third,  and  it  is  accomplished  by  giving 
a  batter  to  the  downstream  face  of  the  dam,  while  the 
upstream  face  still  remains  vertical. 


Joint  ]<&. 

The  investigation  for  the  purpose  of  determining  the 
length  of  /2  involves  ihe  use  of  an  equation  in  which  u 

shall  have  a  value  of'-,  since  the  resultant  of  the  external 
o 

forces  for  the  reservoir  full  reached  the  limit  of  the  middle 
third  at  J\  and  since  it  may  not  pass  outside  that  limit. 
This  is  expressed  by  Eq.  (37).  (Series  A,  Stage  2.) 


As  the  joints  are  taken  every  10  feet  apart,  at  least 
in  the  upper  levels  of  the  dam,  the  factors  in  the  above 
expression  take  the  following  values: 


#  =  52'.  6,     h  =  io't     lo 
=  ii.$,     and     J  =  2.333  (or  J). 


APPENDIX    II  93 

Substituting  these  values  therefore  in  the  above  equa- 
tion, completing  the  square  and  solving  for  /,  we  obtain 

1  =  26.8  feet. 

Now  Eq.  (37)  is  dependent  upon  yQ\  it  is  therefore 
necessary  after  each  application  of  that  equation  to  em- 
ploy Eq.  (36)  in  order  to  solve  for  y,  which  in  turn  be- 
comes y0,  at  the  next  joint.  (See  Supplementary  Equa- 
tions, Stage  II.) 

h 

•A-oyo  i  s  i 

0 
y=- 


As  all  of  these  factors  are  already  known,  it  merely 
requires  that  they  be  substituted  in  the  above. 
This  results  in 


Joint  /3. 

Again  Eq.  (37)  will  be  used  to  determine  the  length 
of  the  joint,  but  the  factors  will  now  have  the  following 
values  : 

^0  =  1689,     H  =  62f.6,     h  =  io',     10  =  26'.8, 

y0  =  n'.6$,    ^  =  2.333. 
These  substituted  in  that  equation  give 
^  =  31.4  feet. 

In  like  manner  we  will  use  Eq.  (36)   to  determine  y. 

This  gives 

=  12.08  feet. 


94  HIGH    MASONRY    DAM    DESIGN 

Joint  J4. 
Using  Eq.  (37)  with  the  following  values: 

^0  =  1980,     h  =  io',     Z0  =  3I'-4>     #  =  72'. 6, 
yo  =  i2  .08,     4  =  2.333. 

Using  Eq.  (36)  to  determine  y,  we  find, 
y  =  12.82  feet. 

Joint  J$. 

While  the  formulae  37  and  36  will  still  be  employed  at 
this  joint  to  determine  the  values  of  /  and  y  respectively, 
h  will  be  taken  equal  to  4.4  feet,  thus  making  #  =  77.0  feet 
.instead  of  82.6  feet.  The  reason  for  this  is  that  if  the 
latter  value  of  H  be  used  the  resulting  value  of  y  wrould 

be  less  than  -,  bringing  the  resultant  for  reservoir  empty 

3 

outside  the  prescribed  limit.     To  keep  it  just  within  this 
limit  it  is  found  by  trial  that  h  should  not  exceed  4.4  feet. 
Using   Eq.    (37)    therefore   to   determine    Z,    with    the 
following  values  inserted, 

^0  =  2321,     fc  =  4'.4,     /0  =  36'-9>     #  =  77'.o, 
?o  =  i2'.82,     ^  =  2.333. 

/  =  39. 5  feet.  ; 

Solving  for  y  in  Eq.  (36), 

^  =  13.2  feet, 
in  which  it  is  seen  that  y  is  practically  J  of  I. 


APPENDIX  II  95 

In  the  employment  of  Eq.  (37)  for  the  determination 
of  the  length  of  the  successive  joints  we  used  the  value 
u  =  ^l,  thus  indicating  that  for  reservoir  full  the  resultant 
acted  at  the  downstream  limit  of  the  middle  third.  On 
the  other  hand,  it  is  now  found  that  for  reservoir  empty 
the  limiting  condition  is  reached,  and  hence  at  this  point 
it  is  necessary  to  batter  the  back  face  while  u  and  y  still 

remain  equal  to  — .     These  conditions  require  the  use  of 

Eq.  (38),  (Series  A,  Stage  III),  for  the  determination  of 
the  length  of  joint,  and  of  Eq.  (44),  Supplimentary  Equa- 
tions, Stage  III,  for  the  determination  of  the  batter  of  the 
upstream  face. 

Joint  Te- 
nsing Eq.  (38), 


with  the  following  values  inserted: 
^0  =  2489,     /*  =  io',     /o  =  39 


whence, 

£  =  48.5  teet. 

To  determine  the  batter,  we  employ  Eq.  (44), 

2A0(l 
6A0 

and  substituting  the  quantities  from  above, 

/  =  i.8  feet. 


96  HIGH    MASONRY    DAM    DESIGN 

In  view  of  the  fact  that  as  the  lower  levels  of  the  dam 
are  approached  the  unit  pressures  increase  and  finally 
become  controlling  factors,  it  will  be  desirable  here  to 
determine  the  intensity  for  the  downstream  edge  of  this 
joint,  where  the  permissible  maximum  normal  unit  stress 
is  only  14  tons  per  square  foot. 

For    this    purpose    Eq.    (14)    will  be    employed,   since 

u=-  and  pf  =  o.     This  application  gives  to  p  the  following 
o 

value  : 

2W 

P=—  7-  =8.  8  tons. 

The  unit  pressure,  it  is  thus  seen,  is  far  below  the  pre- 
scribed limit,  so  that  it  will  be  unnecessary  to  consider 
it  yet  as  a  feature  in  the  design,  although  it  will  be  neces- 
sary to  examine  each  joint  to  determine  at  just  what  point 
it  does  begin  to  control. 

Joint  J7. 

At  this  elevation  we  may  properly  assume  a  depth  h 
of  20  feet  for  the  succeeding  joints,  since  the  larger  value 
will  reduce  the  number  of  applications  of  the  formulae 
and  at  the  same  time  will  not  affect  the  analysis. 

Using  Eq.  (38)  again  with  the  following  values: 

^0  =  2929,     h  =  2o',     /0  =  48/-5,     H 


/  =  64'.6feet. 

Solving  for  t,  the  batter  of  the  upstream  face,  by  Eq. 

(44), 

Z  =  2.    feet. 


APPENDIX  II  97 

Likewise  solving  for  p,  from  Eq.  (14), 
p  =  g.  i  tons. 

Joint  Js. 

Applying  Eqs.  (38),  (44)  and  (14)  for  the  determination 
of  /,  /,  and  p,  respectively,  with  the  following  values  sub- 
stituted : 

^0  =  4060,     h  =  2o',     /0=64/.6,     H  =  i2f.o, 

?o  =  2i'.6,     ^  =  2.333. 
there  results, 

/  =  7p'.7,     /  =  i'.4,     and    p  =  io.i  tons. 

Joint  J9. 

Using  the  same  equations  for  /,  /,  and  p,  with  the  fol- 
lowing values  inserted: 


>     ^  =  20', 


there  results, 


2W 

=  o'.8,   and    P  =  ~T~  =  II-3  tons. 


Joint  JIQ. 

Applying  the  same  formulae  with  the  following  values 
inserted  : 


there  results, 

,     /  =  o'.5,     and    /?  =  i2. 


98  HIGH    MASONRY    DAM    DESIGN 

Joint  Jn. 

Using  the  same  formulae  as  before  with   the  following 
values  inserted: 


yo  =  35-9>     ^  =  2.330, 
/  =  i2i'.2,     /  =  o'.3,     and     £  =  13.  9  tons. 

It  is  noticed  here  that  the  unit  normal  pressure  at 
the  downstream  edge  of  the  joint  has  practically  reached 
the  limit  prescribed  of  14  tons  per  square  foot,  and  it  could 
be  shown  that  an  investigation  of  joint  Ji2,  on  the  same 
lines  as  for  Ju  would  give  a  pressure  at  that  point  con- 
siderably in  excess  of  this  prescribed  value.  It  is  there- 
fore necessary  to  use  such  an  equation  that  this  condition 
of  the  pressure  at  the  downstream  edge  may  be  involved 
in  it. 

This  is  expressed  by  Eq.  (40)   (Series  A,  Stage  IV.) 


Joint  Ji2. 

Using    the    following    values    in    the    above  equation, 
#  =  2o7'.o,  ?  =  62. 5  and  p  =  i$  tons  per  square  foot,  there 

results, 

7  =  140.7  feet. 

To  find  the  batter  of  the  back  we  must  use  Eq.  (44) 
as  heretofore,  with  the  following  values  inserted: 


APPENDIX  II  99 

whence, 

*  =  2'.0. 

It  is  now  also  necessary  to  determine  u,  since  in  the 
above  equation  it  has  no  influence,  and  for  this  purpose 
Eq.  (39)  must  be  used.  (Stage  IV,  Supplementary  Equa- 

tion.) . 

2l       pi2 
-~ 


whence,  substituting  the  values, 

^  =  49.1  feet. 

It  is  also  necessary  to  determine  the  pressure  at  the 
upstream  edge  of  each  joint  since  it  is  gradually  approaching 
the  limit  of  18  tons  per  square  foot. 

Here  Eq.  (14)  will  have  to  be  used, 

2W 
£=—  =  I4.7. 

As  this  value  is  well  inside  the  limit  it  is  unnecessary 
to  use  any  equation  in  which  the  pressure  intensity  at  the 
upstream  face  is  the  controlling  feature. 

Joint  /i3. 

Here  again  we  must  use  Eq.  (40)  with  #  =  227',  £  =  14, 
and  f  =  6  2.  5  whence, 

I  =  i6i.6  feet. 
Solving  for  t, 

t  =  2.i  feet. 
Solving  for  ut 

^  =  59.1  feet. 
Solving  for  g, 

0  =  15.5 


100 


HIGH    MASONRY    DAM    DESIGN 


Investigating  this  last  joint  for  stability  as  to  friction 
we  find, 

H2        2272 
2  2 


=  25765> 


40111, 


H2 


Hence  stability  is  assured. 


Joint 
No. 

H 

H3 

ft 

h  +  a 

At 

A 

lo 

/o2 

I 

42.6 

77.309 

62.6 

1,441 

2 

52.6 

M5.532 

IO 

72.6 

1,441 

1,690 

23.0 

529 

3 

62.6 

245.314 

IO 

82.6 

1,690 

1,982 

26.7 

713 

4 

72.6 

382,657 

IO 

92  .6 

1,982 

2,324 

31.6 

I,OOO 

5 

77-o 

456,533 

4-4 

97.0 

2,324 

2,492 

36.8 

J.356 

6 

87.0 

658,503 

10 

107  .0 

2,492 

2,93° 

39-5 

1,560 

7 

107  .0 

1,225,040 

20 

127  -o 

2,930 

4,059 

48.2 

2,323 

8 

127  .0 

2,048,380 

2O 

147-0 

4,059 

5.503 

64-7 

4,186 

9 

147.0 

3,176,520 

2O 

167  'o 

5.503 

7,240 

79-7 

6,352 

IO 

167  .0 

4,657,460 

20 

187*0 

7,240 

9,255 

94-o 

8,836 

ii 

187.0 

6,539,200 

20 

207   o 

9.255 

11,544 

107.7 

11,449 

12 

207.0 

8,869,740 

20 

227   o 

n,548 

14,169 

121.4 

14,738 

13 

227  .0 

11,697,100 

20 

247.0 

14,169 

17,192 

140.7 

19,800 

Joint 
No. 

I 

yo 

y 

u 

t 

P 

Q 

Aoyo 

7 

I 

23.0 

11  •  5 

2 

26.7 

ii  .5 

ii  .64 

1 

16,560 

3 

31-6 

ii  .64 

12  .08 

i 

19,653 

4 

36.8 

12.08 

12.  80 

i 

23.913 

5 

39-5 

12.80 

13  .23 

T_ 

9.1 

29.773 

6 

48.2 

13  .23 

16.07 

7 
3 

1.64 

8.9 

7 

64-7 

16.1 

21  .  6 

1 

2.4 

9-2 

8 

79-7 

21.6 

26.6 

1 

I  .  2 

IO.  I 

9 

94-o 

26.6 

3r-3 

J7 

0.8 

I  I  .  2 

IO 

107.7 

31  .3 

35-9 

7 
~S 

0.5 

12.6 

ii 

121  .4 

35-9 

40.4 

1 

0.8 

13-9 

12 

140.7 

40.4 

46.9 

49.0 

2  .  0 

I4.O 

13 

161.6 

I 

3 

I 

3 

59-2 

2.9 

14-0 

15-5 

DATA 

"Wt.  of  Masonry  =  146  /cu.  ft. 
>Wt.  of  Water  =62.5 */cu.  ft. 
p  =  14  Tons 
=  18  Tons 


Bocfe 


FIG  17. 


101 


Inspection 

Galle 
Flow  Line  E1.590 


Concrete  drainage  > 
blocks 


El  .500 
Drainage  Well 


FIG.  18. — Olive  Bridge  Dam. 


102 


j  Drainage  Well 

CROSS-SECTION  AT  TOP 

2         0        2       4        6        8       10ft. 

2M. 


0 


FIG.  19. 


103 


201338 


, 


